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 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.