Searches for the exclusive decays of Higgs and $Z$ bosons into a vector quarkonium state and a photon are performed in the $\mu^+\mu^- \gamma$ final state with a proton--proton collision data sample corresponding to an integrated luminosity of $139$ fb$^{-1}$ collected at $\sqrt{s}=13$ TeV with the ATLAS detector at the CERN Large Hadron Collider. The observed data are compatible with the expected backgrounds. The 95% CL$_\mathrm{s}$ upper limits on the branching fractions of the Higgs boson decays into $J/\psi \gamma$, $\psi(2S) \gamma$, and $\Upsilon(1S,2S,3S) \gamma$ are found to be $2.1\times10^{-4}$, $10.9\times10^{-4}$, and $(2.6,4.4,3.5)\times10^{-4}$, respectively, assuming Standard Model production of the Higgs boson. The corresponding 95% CL$_\mathrm{s}$ upper limits on the branching fractions of the $Z$ boson decays are $1.2\times10^{-6}$, $2.3\times10^{-6}$, and $(1.0,1.2,2.3)\times10^{-6}$.

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