A search is presented for decays of Z and Higgs bosons to a J/$\psi$ meson and a photon, with the subsequent decay of the J/$\psi$ to $\mu\mu$. The analysis uses data from proton-proton collisions with an integrated luminosity of 35.9 fb$^{-1}$ at $\sqrt{s}$ = 13 TeV collected with the CMS detector at the LHC. The observed limit on the $Z\rightarrow J/\psi\gamma$ decay branching fraction, assuming that the J/$\psi$ meson is produced unpolarized, is $1.4\times10^{-6}$ at 95% confidence level, which corresponds to a rate higher than expected in the standard model by a factor of 15. For extreme-polarization scenarios, the observed limit changes from -13.6 to +8.6% with respect to the unpolarized scenario. The observed upper limit on the branching fraction for $H\rightarrow J/\psi\gamma$ where the J/$\psi$ meson is assumed to be transversely polarized is $7.6\times 10^{-4}$, a factor of 260 larger than the standard model prediction. The results for the Higgs boson are combined with previous data from proton-proton collisions at $\sqrt{s}$ = 8 TeV to produce an observed upper limit on the branching fraction for $H\rightarrow J/\psi\gamma$ that is a factor of 220 larger than the standard model value.