A search is presented for decays of the Higgs and $Z$ boson to pairs of $J/\psi$ or $\Upsilon(nS)$ ($n=1,2,3)$, with the subsequent decay of the mesons to $\mu^+\mu^-$. The analysis uses data from proton-proton collisions with an integrated luminosity of 37.5~\mathrm{fb}^{-1} at \sqrt{s}=13~\mathrm{TeV} collected with the CMS detector at the LHC in 2017. An observation of such a decay with this sample would indicate the presence of physics beyond the standard model. No significant excess of events is observed and $95\%$ confidence level upper limits are placed on the branching fractions. In the $J/\psi$ pair final state the limits are $1.8\times 10^{-3}$ and $2.2\times 10^{-6}$ for the Higgs and $Z$ boson, respectively, while in the combined $\Upsilon(nS)$ pair final states the limits are $1.4\times 10^{-3}$ and $1.5\times 10^{-6}$, respectively. The $J/\psi$ and $\Upsilon(nS)$ mesons from the Higgs and $Z$ decay are assumed unpolarized. With longitudinally polarized $J/\psi$ and $\Upsilon$ mesons in the Higgs and $Z$ decay the upper limits reduce by about $22-29\%$, while with transversely polarized mesons the decay limits increase by about $10-13\%$.

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