Searches for the exclusive decays of the Higgs and Z bosons into a J/ψ, ψ(2S), or Υ(nS) (n = 1, 2, 3) meson and a photon are performed with a pp collision data sample corresponding to an integrated luminosity of 36.1 fb−1 collected at √s = 13 TeV with the ATLAS detector at the CERN Large Hadron Collider. No significant excess of events is observed above the expected backgrounds, and 95% confidence- level upper limits on the branching fractions of the Higgs boson decays to J/ψ γ , ψ(2S) γ , and Υ(nS) γ of 3.5 × 10−4, 2.0 × 10−3, and (4.9, 5.9, 5.7) × 10−4, respectively, are obtained assuming Standard Model production. The corresponding 95% confidence-level upper limits for the branching fractions of the Z boson decays are 2.3 × 10−6, 4.5 × 10−6 and (2.8, 1.7, 4.8) × 10−6, respectively.