A search for decays of the Higgs and Z boson to pairs of J/ψ or Y(nS) (n = 1, 2, 3) mesons, with their subsequent decay to μ+μ− pairs, is presented. The analysis uses data from proton-proton collisions at s=13TeV, collected with the CMS detector at the LHC in 2017 and corresponding to an integrated luminosity of 37.5fb−1. While an observation of such a decay with this sample would indicate the presence of physics beyond the standard model, no significant excess is observed. Upper limits at 95% confidence level are placed on the branching fractions of these decays. In the J/ψ pair channel, the limits are 1.8×10−3 and 2.2×10−6 for the Higgs and Z boson, respectively, while in the combined Y(nS) pair channel, the limits are 1.4×10−3 and 1.5×10−6, respectively, when the mesons from the Higgs and Z boson decay are assumed to be unpolarized. When fully longitudinal and transverse polarizations are considered the limits reduce by about 22–29% and increase by about 10–13%, respectively.