Inclusion Criteria:  Have a diagnosis of psoriatic arthritis (PsA) for greater than or equal to (>=) 6 months prior to the first administration of study intervention and meet Classification criteria for PsA (CASPAR) criteria at screening Have active PsA as defined by: at least 3 swollen joints and at least 3 tender joints at screening and at baseline; and high-sensitivity C-reactive protein (hsCRP) >= 0.3 milligrams per deciliter (mg/dL) at screening from the central laboratory Have at least 1 of the following PsA subsets: distal interphalangeal joint involvement, polyarticular arthritis with absence of rheumatoid nodules, arthritis mutilans, asymmetric peripheral arthritis, or spondylitis with peripheral arthritis Have active plaque psoriasis, with at least one psoriatic plaque of >=2 centimeter (cm) diameter or nail changes consistent with psoriasis Have an inadequate response (IR) to 1 anti-tumor necrosis factor-alpha (anti-TNF-alpha) therapy, defined as presence of active PsA despite previous treatment with only 1 prior anti-TNF-alpha agent and the following: a. Lack of benefit to 1 prior anti-TNF-alpha therapy, as documented in the participant history by the treating physician, after at least 12 weeks of etanercept, adalimumab, certolizumab pegol therapy, or at least 14-weeks of infliximab, or any biosimilar of these 4 therapies. Documented lack of benefit may include inadequate improvement in joint counts, physical function, or disease activity; b. The last dose of anti-TNF-alpha therapy must have occurred greater than 5 half-lives of the drug prior to screening visit (washout period)  Exclusion Criteria:  Has other inflammatory diseases that might confound the evaluations of benefit of guselkumab and/or golimumab therapy, including but not limited to rheumatoid arthritis (RA), ankylosing spondylitis (AS), nonradiographic axial spondyloarthritis (nr AxSpA), systemic lupus erythematosus, or lyme disease Has known intolerance or hypersensitivity to any biologic medication, or known allergies or clinically significant reactions to murine, chimeric, or human proteins, monoclonal antibodies (mAb), or antibody fragments Has received prior treatment with golimumab or guselkumab or has documented intolerance to prior anti-TNF-alpha therapy in the participant history by the treating physician Has received more than 1 prior anti-TNF-alpha agent (or biosimilars) Positive human immunodeficiency virus (HIV) antibody test