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 36.1fb−1 collected at s=13TeV 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/ψγ , 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.