Study of Safety and Tolerability of Nivolumab Treatment Alone or in Combination With Relatlimab or Ipilimumab in Head and Neck Cancer

Inclusion Criteria: 1. Males and females, ages ≥18 years 2. Histologically or cytologically confirmed Squamous Cell Carcinoma, previously untreated stage III, or IVA HNC by AJCC 8th edition staging system. Newly diagnosed, never treated HNC cancer but could have had a surgically treated primary \> 5 years previous without radiotherapy or chemotherapy. For HPV positive oropharyngeal cancer, patients with T3 or T4 primary and/or one ipsilateral lymph node greater than 3 cm, multiple ipsilateral lymph nodes, bilateral lymph nodes, or contralateral lymph node will be included. Patients must undergo CT or MRI to rule out the presence of distant metastases. 3. Accessible tumor for pretreatment (baseline) open/incisional biopsy to provide adequate correlative specimen. 4. Have LAG-3 and PD-L1 results for stratification. 5. LVEF assessment with documented LVEF ≥50% by either TTE or MUGA (TTE preferred test) within 28 days prior to first study drug administration 6. Women of child-bearing potential (WOCBP) must be advised of the importance of avoiding pregnancy during trial participation and the potential risk factors for an unintentional pregnancy. All WOCBP MUST have a negative pregnancy test within 7 days prior to first receiving investigational product. If the pregnancy test is positive, the patient must not receive investigational product and must not be enrolled in the study. All WOCBP must agree to use appropriate contraception to prevent pregnancy for the duration of treatment with study treatments, plus 24 weeks after the last dose of study treatment (i.e., 30 days \[duration of ovulatory cycle\] plus approximately 5 half-lives). 7. All males must agree to use appropriate contraception for the duration of treatment with study treatments plus 33 weeks after the last dose of study treatment (i.e., 90 days \[duration of sperm turnover\] plus approximately 5 half-lives). In addition, male participants must be willing to refrain from sperm donation during this time. In addition, men enrolled on this study must be informed of the risks to any sexual partner of childbearing potential and should practice an effective method of birth control 8. Azoospermic males are exempt from contraceptive requirements unless the potential exists for fetal toxicity due to study drug being present in seminal fluid, even if the participant has undergone a successful vasectomy or if the partner is pregnant. WOCBP who are continuously not heterosexually active are also exempt from contraceptive requirements, and still must undergo pregnancy testing as described in this section. 9. Primary tumors of the oral cavity, oropharynx, hypopharynx, or larynx will be included. 10. Eligible for surgical resection. 11. Age ≥ 18 years 12. ECOG performance status 0-1. 13. Have signed written informed consent Exclusion Criteria: 1. Prior radiation, chemotherapy, oncology vaccine or immunotherapy. 2. Prior severe infusion reaction to a monoclonal antibody. 3. Troponin T (TnT) or I (TnI) \> 2 × institutional ULN. Subjects with TnT or TnI levels between \> 1 to 2 × ULN will be permitted if repeat levels within 24 hours are less than or equal to 1 x ULN. If TnT or TnI levels are \> 1 to 2 × ULN within 24 hours, the subject may undergo a cardiac evaluation and be considered for treatment, following a discussion with the BMS Medical Monitor or designee. When repeat levels within 24 hours are not available, a repeat test should be conducted as soon as possible. If TnT or TnI repeat levels beyond 24 hours are \< 2 x ULN, the subject may undergo a cardiac evaluation and be considered for treatment. 4. Evidence of distant metastasis. 5. Prior history of HNC treated \< 5 years previously. 6. Prior history of myocarditis, regardless of etiology 7. Prior treatment with LAG-3 targeted agents. 8. A known history of Hepatitis B or C 9. Patients with active/history of autoimmune disease. "Active" refers to any condition currently requiring therapy. Examples of autoimmune disease include systemic lupus erythematosus, multiple sclerosis, inflammatory bowel disease and rheumatoid arthritis 10. Psychiatric illness or other social issues limiting compliance 11. If second primary tumor is found at the time of EUA, the subject will be excluded from study participation.