A search for the exclusive decays of the Higgs and Z bosons to a ϕ or ρ meson and a photon is performed with a pp collision data sample corresponding to an integrated luminosity of up to 35 . 6 fb −1 collected at √s = 13 TeV with the ATLAS detector at the CERN Large Hadron Collider. These decays have been suggested as a probe of the Higgs boson couplings to light quarks. No significant excess of events is observed above the background, as expected from the Standard Model. Upper limits at 95% confidence level were obtained on the branching fractions of the Higgs boson decays to ϕ γ and ρ γ of 4 . 8 × 10 −4 and 8 . 8 × 10 −4 , respectively. The corresponding 95% confidence level upper limits for the Z boson decays are 0 . 9 × 10 −6 and 25 × 10 −6 for ϕ γ and ρ γ, respectively.