A Phase II Trial of Neoadjuvant Treatment With PD-1 Inhibition (Nivolumab) With or Without IDO Inhibition (BMS-986205) and With or Without CTLA-4 Inhibition (Ipilimumab) in Resectable Stage III or IV Melanoma

Inclusion Criteria: * Histologic or cytologic diagnosis of resectable stage III or IV cutaneous melanoma. Patients with melanoma of mucosal origin are not eligible. Patients with acral melanoma that fit criteria are eligible. Patients must have clinically detectable stage III (clinically detectable N1b, N1c, N2b, N2c, N3b or N3c) or stage IV resectable melanoma. * Age ≥ 18 years. * ECOG performance status ≤2 (Karnofsky ≥60%, see Appendix A) * Participants must have normal organ and marrow function as defined below: * leukocytes ≥2,000/mcL * absolute neutrophil count ≥1,500/mcL * platelets ≥100,000/mcL * hemoglobin ≥9.0g/dL * total bilirubin within normal institutional limits * AST(SGOT)/ALT(SGPT) ≤2.5 × institutional upper limit of normal * creatinine within normal institutional limits OR * creatinine clearance ≥40 mL/min/1.73 m2 for participants with creatinine levels above institutional normal. * C-Reactive Protein \< institutional ULN * Quantitative or qualitative G6PD assay results must not suggest underlying G6PD deficiency * Measurable disease (by CT, PET/CT or MRI) * Prior therapies including targeted therapy and immunotherapy are not allowed, with the exception of adjuvant Ipilimumab or Interferon-α-2b. * Ability to understand and the willingness to sign a written informed consent document. * Ability to swallow pills intact and without GI issues which may impact medication absorption. * Women of childbearing potential (WOCBP) must agree to follow instructions for method(s) of contraception (Refer to Appendix B) for the duration of treatment with study treatment(s) plus 5 months post-treatment completion (i.e. 30 days plus the time required for nivolumab to undergo approximately 5 half-lives). * Women of childbearing potential (WOCBP) must have a negative serum or urine pregnancy test (minimum sensitivity 25 IU/L or equivalent units of HCG) within 24 hours prior to the start of study treatment. * Males who are sexually active with WOCBP must agree to follow instructions for method(s) of contraception for the duration of treatment with study treatment(s) plus 7 months post-treatment completion (i.e. 90 days plus the time required for nivolumab to undergo approximately 5 half-lives). In addition, male participants must be willing to refrain from sperm donation during this time. This criterion applies to azoospermic males as well. Exclusion Criteria: * A history of prior treatment with PD-1 inhibitor, CTLA-4 inhibitor or IDO inhibition. Prior therapy with ipilimumab or Interferon-α-2b in the adjuvant setting is permitted. Participants may not have received live/attenuated vaccines within 30 days of first treatment. * Participants with uveal or mucosal melanoma * Participants with known brain metastases must have documented stability for at least 30 days directly prior to study enrollment and not be requiring active treatment for these. Prior radiation, surgery and stereotactic radiosurgery are allowed but must be completed four weeks prior to initiating therapy. * History of allergic reactions attributed to compounds of similar chemical or biologic composition to nivolumab, ipilimumab, methylene blue or BMS-986205. History or presence of hypersensitivity or idiosyncratic reaction to methylene blue. * Need for systemic steroids at the time of enrollment. Physiologic replacements at a dose of less than 10 mg daily prednisone equivalent is allowed. * Blood Methemoglobin \> ULN, assessed in an arterial or venous blood sample or by co-oximetry * Participants with active ILD/pneumonitis or history of ILD/ pneumonitis requiring steroids. * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements. * Pregnant women are excluded from this study because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with nivolumab or BMS-986205, breastfeeding should be discontinued if the mother is treated with nivolumab or BMS-986205. These potential risks may also apply to other agents used in this study. * Known active HIV, Hepatitis B or Hepatitis C patients. HIV-positive participants on combination antiretroviral therapy are ineligible because of the potential for an immunologic effect with the therapy. Appropriate studies will be undertaken in participants receiving combination antiretroviral therapy when indicated. * Autoimmune disease that requires treatment at the time of enrollment. Participants with type I diabetes mellitus, hypothyroidism only requiring hormone replacement, skin disorders (such as vitiligo, psoriasis, or alopecia) not requiring systemic treatment or conditions not expected to recur in the absence of an external trigger are permitted to enroll. * Participant with a personal or family (ie, in a first-degree relative) history of cytochrome b5 reductase deficiency (previously called methemoglobin reductase deficiency) or other diseases that put them at risk of methemoglobinemia. All participants will be screened for methemoglobin levels prior to randomization. * Participant with a history of G6PD deficiency or other congenital or autoimmune hemolytic disorders. All participants will be screened for G6PD levels prior to randomization. * Participants must not have a history of life-threatening toxicity related to prior immune therapy (eg. anti-CTLA-4 or any other antibody or drug specifically targeting T-cell co-stimulation or immune checkpoint pathways) except those that are unlikely to re-occur with standard countermeasures (eg. hormone replacement after adrenal crisis). * Treatment with botanical preparations (eg, herbal supplements or traditional Chinese medicines) intended for general health support or to treat the disease under study within 2 weeks prior to randomization. * History of other malignancy within 3 years prior to screening, with the exception of those with a negligible risk of metastasis or death (e.g., 5-year OS of \> 90%), such as adequately treated carcinoma in situ of the cervix, non-melanoma skin carcinoma, localized prostate cancer, ductal carcinoma in situ, or Stage I uterine cancer. * Participants who have had major surgery requiring general anesthesia or significant trauma who have not recovered per physician determination for at least 14 days prior to randomization. * Participants with conditions known to interfere significantly with the absorption of oral medication, as per investigator judgment. * Participants with uncontrolled adrenal insufficiency. * Prior history of serotonin syndrome.