A search for the exclusive decays of the Higgs and Z bosons to a J/ψ, ψ(2S), Υ(1S), Υ(2S), or Υ(3S) meson and a photon is performed with a proton-proton collision data sample corresponding to an integrated luminosity of 139 fb−1 collected at a center of mass energy of √s = 13 TeV with the ATLAS detector at the CERN Large Hadron Collider. The decays H/Z→Mγ→μ+μ−γ are considered in two different selection channels of 2.4 GeV/c2< mμ+μ−<4.3 GeV/c2and 8.0 GeV/c2< mμ+μ−<12.0 GeV/c2 for the ψ(nS) and the Υ(nS) respectively. The H signal is modelled with Monte-Carlo (MC) simulated events that correspond to the primary proton-proton production mechanisms, including; gluon fusion, vector boson fusion, vector boson associated production and top quark associated production. The total background is made up of exclusive background Drell-Yan Z→μ+μ−γ MC events and an inclusive background model made from a non-parametric data driven approach. The blinded post-fit expected 95% confidence level upper limits on the branching fractions of the Higgs decays toψ(nS) and Υ(nS) of (2.3,7.0)×10−4 and (2.2,2.6,2.3)×10−4, respectively are obtained. The corresponding 95% confidence level upper limits on the branching fractions of the Z decays toψ(nS) and Υ(nS) are (0.67,3.0)×10−6 and (1.5,1.9,1.6)×10−6, respectively.