A search for the decays of the Higgs and $Z$ bosons to $J/\ensuremath{\psi}\ensuremath{\gamma}$ and $\mathrm{\ensuremath{\Upsilon}}(nS)\ensuremath{\gamma}$ ($n=1,2,3$) is performed with $pp$ collision data samples corresponding to integrated luminosities of up to $20.3\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ collected at $\sqrt{s}=8\text{ }\text{ }\mathrm{TeV}$ with the ATLAS detector at the CERN Large Hadron Collider. No significant excess of events is observed above expected backgrounds and 95% C.L. upper limits are placed on the branching fractions. In the $J/\ensuremath{\psi}\ensuremath{\gamma}$ final state the limits are $1.5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ and $2.6\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ for the Higgs and $Z$ boson decays, respectively, while in the $\mathrm{\ensuremath{\Upsilon}}(1S,2S,3S)\ensuremath{\gamma}$ final states the limits are $(1.3,1.9,1.3)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ and $(3.4,6.5,5.4)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$, respectively.