Searches for the exclusive decays of Higgs and Z bosons into a vector quarkonium state and a photon are performed in the μ+μ-γ final state with a proton–proton collision data sample corresponding to an integrated luminosity of 139 fb ^{- 1} collected at s=13 TeV with the ATLAS detector at the CERN Large Hadron Collider. The observed data are compatible with the expected backgrounds. The 95% confidence-level upper limits on the branching fractions of the Higgs boson decays into J/ψγ , ψ(2S)γ , and Υ(1S,2S,3S)γ are found to be 2.0 × 10 ^{- 4} , 10.5 × 10 ^{- 4} , and (2.5 , 4.2 , 3.4) × 10 ^{- 4} , respectively, assuming Standard Model production of the Higgs boson. The corresponding 95% CL upper limits on the branching fractions of the Z boson decays are 1.2 × 10 ^{- 6} , 2.4 × 10 ^{- 6} , and (1.1 , 1.3 , 2.4) × 10 ^{- 6} . An observed 95% CL interval of (- 133 , 175) is obtained for the κ_c/ κ_γ ratio of Higgs boson coupling modifiers, and a 95% CL interval of (- 37 , 40) is obtained for κ_b/ κ_γ .