Inclusion Criteria:

- STEP 1 - SPECIMEN SUBMISSION INCLUSION CRITERIA

- Participants must have histologically confirmed melanoma that is stage III or IV,
unresectable, recurrent, or metastatic non-uveal melanoma OR Participants must have
histologically confirmed squamous cell carcinoma of the head and neck (HNSCC) that is
either locally recurrent and non-amendable to curative therapy (e.g., radiation,
surgery) or metastatic. The primary tumor location must be the oropharynx, oral
cavity, hypopharynx, or larynx. Primary tumor site of nasopharynx (any histology) or
unknown primary tumor are not eligible.

- Note: For participants with primary oropharyngeal cancer, human papillomavirus
(HPV) or p16 status must be known prior to step 1 registration

- Participants must have disease presentation consistent with measurable disease. Note:
Current disease measurements will not be required until step 2 registration.

- Participants must have had documented progression within 12 weeks after the last dose
of PD-1 checkpoint inhibition-based therapy. Participants must have been receiving
checkpoint inhibition for a minimum of 6 weeks prior to progression. Participants who
recur during adjuvant anti-PD1 treatment or within 12 weeks of completion of adjuvant
anti-PD1 treatment are eligible if they have measurable disease and considered
unresectable.

- Participants with known human immunodeficiency virus (HIV)-infection must be receiving
anti-retroviral therapy and have an undetectable viral load test within 6 months prior
to step 1 registration

- Participants with evidence of chronic hepatitis B virus (HBV) infection must have
undetectable HBV viral load within 28 days prior to step 1 registration.

- Participants with a history of hepatitis C virus (HCV) infection must have no
detectable viral load within 28 days prior to step 1 registration.

- Participants must have recovered to baseline or ≤ Grade 1 Common Terminology Criteria
for Adverse Events (CTCAE) version (v) 5 toxicities related to any prior treatments,
unless adverse events are deemed clinically nonsignificant by the treating
investigator or stable on supportive therapy.

- Participants must be ≥ 18 years of age.

- Participants must have a Zubrod Performance Status 0 or 1.

- Participants must have adequate cardiac function. Participants with known history or
current symptoms of cardiac disease, or history of treatment with cardiotoxic agents,
must have a clinical risk assessment of cardiac function using the New York Heart
Association Functional Classification and must be class 2B or better to be eligible
for this trial.

- Have adequate tissue specimen from procedure obtained within 6 months prior to step 1
registration and after development of resistance to anti-PD-1/L1 therapy. Archival
tissue must consist of tumor block or at least 20 freshly cut serially sectioned and
numbered 4-5 micron unstained, positively-uncharged slides OR

Be willing to undergo research biopsy AND have tumor accessible for biopsy based on the
following criteria:

- Mediastinal, laparoscopic, gastrointestinal, or bronchial endoscopic biopsies can be
obtained incidentally to a clinically necessary procedure and NOT for the sole purpose
of the clinical trial.

- Acceptable biopsy procedures are:

- Percutaneous biopsy with local anesthetic and/or sedation with an expected risk
of severe complications < 2%;

- Direct transoral biopsy (with or without local anesthetic and/or sedation) with
an expected risk of severe complications < 2%;

- Excisional cutaneous biopsy with local anesthetic and/or sedation with an
expected risk of severe complications < 2%;

- Biopsy with removal of additional tumor tissue during a medically necessary
mediastinoscopy, laparoscopy, gastrointestinal endoscopy, bronchoscopy or
craniotomy. No open surgical, laparoscopic or endoscopic procedure should be
performed solely to obtain a biopsy for this protocol;

- Removal of additional tumor tissue during a medically necessary surgical
procedure.

- Participants must have been offered the opportunity to participate in
specimen banking.

- Note: As a part of the Oncology Patient Enrollment Network (OPEN)
registration process the treating institution's identity is provided in
order to ensure that the current (within 365 days) date of institutional
review board approval for this study has been entered in the system.

- Participants must be informed of the investigational nature of this study and must
sign and give informed consent in accordance with institutional and federal guidelines.

- Participants with impaired decision-making capacity are eligible as long as their
neurological or psychological condition does not preclude their safe participation in
the study (e.g., tracking pill consumption and reporting adverse events to the
investigator).

- STEP 2 TREATMENT REGISTRATION INCLUSION CRITERIA

- Participants must have been eligible for step 1 registration.

- Participants must have had their tumor tissue submitted via the Southwest
Oncology Group (SWOG) Specimen Tracking System prior to step 2 registration.

- Participants registered during stage II of the protocol must have received
assignment to an open cohort from the SWOG Statistics and Data Management Center
based on their biomarker screening profile (not applicable for patients
registered during stage I of the protocol).

- Participants must have measurable disease. All measurable disease must be
assessed within 28 days prior to step 2 registration. All non-measurable disease
must be assessed within 42 days prior to step 2 registration. Note: All disease
must be assessed and documented on the Baseline Tumor Assessment Form (Response
Evaluation Criteria in Solid Tumors [RECIST] 1.1).

- For melanoma participants, computed tomography (CT) chest, abdomen and pelvis
must be obtained. For HNSCC participants, CT neck and chest must be obtained.
Further imaging (i.e., magnetic resonance imaging [MR] brain, CT abdomen/pelvis
or extremities, bone scan) will be performed as deemed appropriate by the
treating physician.

- Participants with treated brain metastases must have no evidence of progression
on the follow-up brain imaging after central nervous system (CNS)-directed
therapy.

- Participants with treated brain metastases must have discontinued steroid
treatment at least 14 days prior to step 2 registration.

- Participants must have a history and physical examination performed within 28
days prior to step 2 registration.

- Participants must be able to take oral medication without breaking, opening,
crushing, dissolving or chewing capsules.

- Leukocytes ≥ 3,000/uL (within 28 days prior to step 2 registration).

- Absolute neutrophil count ≥ 1,500/uL (within 28 days prior to step 2
registration).

- Platelets ≥ 100,000/uL (within 28 days prior to step 2 registration).

- Total bilirubin ≤ 1.5 x institutional upper limit of normal (ULN) or ≤ 3 x ULN
for participants with Gilbert's disease (within 28 days prior to step 2
registration).

- Aspartate aminotransferase (AST) ≤ 3 x institutional ULN (within 28 days prior
to step 2 registration).

- Alanine aminotransferase (ALT) ≤ 3 x institutional ULN (within 28 days prior to
step 2 registration).

- Urinalysis: For baseline value (no required value for eligibility).

- Fridericia corrected QT interval (QTcF) ≤ 470 milliseconds on screening
electrocardiogram (ECG) (within 28 days prior to step 2 registration).

- Creatinine ≤ 3 x institutional ULN OR measured (OR calculated) creatinine
clearance ≥ 30 mL/min using the following Cockcroft-Gault Formula. This specimen
must have been drawn and processed within 28 days prior to Step 2 registration.

Exclusion Criteria:

- STEP 1 - SPECIMEN SUBMISSION EXCLUSION CRITERIA

- Participants must not have an active infection requiring systemic therapy (except HBV,
HCV or HIV as mentioned above).

- Participants must not have received surgery, chemotherapy, radiation therapy, biologic
agents, or steroids within 14 days prior to step 1 registration.

- Participants must not have received more than one prior primary radiotherapy regimen,
curative or adjuvant, to the head and neck, with the additional following criteria:

- If the primary radiation is combined with chemotherapy, a minimum of 16 weeks
will be required to have elapsed between the end of radiotherapy and step 1
registration. If the radiation is given alone, a minimum of 8 weeks will be
required to have elapsed between the end of radiotherapy and step 1 registration;

- Additional palliative radiotherapy regimens are permitted but cannot have been
administered to previously treated tissue (i.e., overlapping fields are excluded)
and must be completed at least 4 weeks prior to step 1 registration;

- Treatment areas should be healed with no sequelae from radiation therapy (RT)
that would predispose to fistula formation;

- Participants must not have received prior treatment with anti-VEGF therapies for any
reason;

- Participants must not have any known significant organ disfunction that, in the
opinion of the treating investigator, may impact suitability for receiving combination
nivolumab/cabozantinib treatment.

- Participants must not have active autoimmune disease requiring systemic steroids
(equivalent of > 10mg of prednisone) or other immune suppression. Exceptions:

- Type 1 diabetes mellitus;

- Hypothyroidism only requiring hormone replacement;

- Skin disorders (e.g., vitiligo, psoriasis, or alopecia) not requiring systemic
treatment;

- Conditions not expected to recur in the absence of an external trigger.

- Participants must not have received an organ allograft.

- Participants must not have a history of hemoptysis (defined as ≥ 1/2 tsp of bright
red blood per day) or tumor bleeding within 90 days prior to step 1 registration.

- Participants must not have known central lung lesions involving major blood vessels
(arteries or veins) or a tumor encasing major blood vessels (e.g., carotid artery).

- Participants must not require concomitant anticoagulation with coumarin agents (e.g.,
warfarin), direct thrombin inhibitors (e.g., dabigatran), direct factor Xa inhibitor
betrixaban, or platelet inhibitors (e.g., clopidogrel).

- Participants must not require anticoagulants except for the following:

- Prophylactic use of low-dose aspirin for cardio-protection (per local
applicable guidelines) and low-dose low molecular weight heparins (LMWH);

- Therapeutic doses of LMWH or anticoagulation with direct factor Xa
inhibitors, rivaroxaban, edoxaban, or apixaban in participants without known
brain metastases who are on a stable dose of the anticoagulant for at least
1 week prior to step 1 registration without clinically significant
hemorrhagic complications from the anticoagulation regimen or the tumor.

- Participants must not have evidence of preexisting uncontrolled hypertension prior to
step 1 registration as documented by 2 baseline blood pressure readings taken at least
30 minutes apart within 28 days prior to step 1 registration. The baseline systolic
blood pressure readings must be ≤ 150 mmHg and the baseline diastolic blood pressure
readings must be ≤ 90 mmHg. Participants on antihypertensive therapies with
controlled blood pressure are eligible.

- Participants must not have a prior or concurrent malignancy whose natural history or
treatment (in the opinion of the treating physician) has the potential to interfere
with the safety or efficacy assessment of the investigational regimen.

- Participants must not be pregnant or nursing due to the known safety profiles of the
drugs in this study. Individuals who are of reproductive potential must have agreed to
use an effective contraceptive method with details provided as a part of the consent
process. A person who has had menses at any time in the preceding 12 consecutive
months or who has semen likely to contain sperm is considered to be of "reproductive
potential. In addition to routine contraceptive methods, "effective contraception"
also includes refraining from sexual activity that might result in pregnancy and
surgery intended to prevent pregnancy (or with a side-effect of pregnancy prevention)
including hysterectomy, bilateral oophorectomy, bilateral tubal ligation/occlusion and
vasectomy with testing showing no sperm in the semen.

- STEP 2 TREATMENT REGISTRATION EXCLUSION CRITERIA

- Participants must not have experienced any significant health changes that, in the
opinion of the treating investigator, may impact continued suitability for receiving
combination nivolumab/cabozantinib treatment.

- Participants must not have received more than one prior primary radiotherapy regimen,
curative or adjuvant, to the head and neck, with the additional following conditions:

- If the primary radiation is combined with chemotherapy, a minimum of 16 weeks
will be required to have elapsed between the end of radiotherapy and step 2
registration. If the radiation is given alone, a minimum of 8 weeks will be
required to have elapsed between the end of radiotherapy and step 2 registration;

- Additional palliative radiotherapy regimens are permitted but cannot have been
administered to previously treated tissue (i.e., overlapping fields are excluded)
and must be completed at least 4 weeks prior to step 2 registration.

- Treatment areas should be healed with no sequelae from RT that would predispose
to fistula formation.

- Participants must not have received investigational agents or monoclonal antibodies
(except Food and Drug Administration [FDA] approved supportive care antibodies, such
as denosumab) within 28 days prior to step 2 registration.

- Participants must not have received any prior treatment with anti-VEGF based therapies.

- Participants must not have received administration of a live, attenuated vaccine
within 30 days prior to step 2 registration. Note: Participants may have received a
messenger ribonucleic acid (mRNA) or viral vector-based COVID-19 vaccine within 30
days prior to step 2 registration.

- Participants must not have received administration of any strong CYP3A4 inducers, such
as but not limited to rifampin, carbamazepine, enzalutamide, mitotane, phenytoin and
St. John's wort, within 14 days prior to step 2 registration.

- Participants must not have received administration of any strong CYP3A4 inhibitors,
such as but not limited to clarithromycin, itraconazole, ketoconazole, grapefruit
juice, indinavir, nelfinavir, ritonavir, nefazodone, saquinavir, and telithromycin,
within 5 times the half-life of the CYP3A inhibitor prior to step 2 registration.

- Participants must not have malabsorption syndrome.

- Participants must not have central lung lesions involving major blood vessels
(arteries or veins) or a tumor encasing major blood vessels (e.g., carotid artery).

Note: Other protocol defined Inclusion/Exclusion Criteria may apply.

Eligibility last updated 11/21/22. Questions regarding updates should be directed to the study team contact.