A search for the decays of the Higgs and Z bosons to J/ψγ and ϒ(nS)γ (n=1,2,3) is performed with pp collision data samples corresponding to integrated luminosities of up to 20.3 fb^{-1} collected at sqrt[s]=8 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/ψγ final state the limits are 1.5×10^{-3} and 2.6×10^{-6} for the Higgs and Z boson decays, respectively, while in the ϒ(1S,2S,3S)γ final states the limits are (1.3,1.9,1.3)×10^{-3} and (3.4,6.5,5.4)×10^{-6}, respectively.