A search for the exclusive decays of the Higgs and Z bosons to a phi or rho 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 root 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 phi gamma and rho gamma of 4.8 x 10(-4) and 8.8 x 10(-4), respectively. The corresponding 95% confidence level upper limits for the Z boson decays are 0.9 x 10(-6) and 25 x 10(-6) for phi gamma and rho gamma, respectively.