For the first time, decays of the Higgs boson into a $\mathrm{Z}$ boson and a $\mathrm{J/}\psi$ meson are searched for in four-lepton final states with the CMS detector at the LHC. A proton-proton collision data set corresponding to an integrated luminosity of 138~\mathrm{fb}^{-1} is exploited. In addition, Higgs decays into pairs of $\mathrm{J/}\psi$ or $\Upsilon$ mesons are studied in the same data. Events with $\mathrm{Z}$ boson decays into an electron or muon pair, or with quarkonium resonances decaying into muon pairs, are selected. An observation of such decays with this sample would indicate the presence of physics beyond the standard model, and no significant excess is observed. Upper limits at the $95\%$ confidence level are placed on the branching fractions of these decays. Higgs decay branching fractions for \mathrm{H}\to \mathrm{Z}~\psi\text{(2S)}, \mathrm{H}\to \mathrm{J/}\psi~\psi\text{(2S)}, and \mathrm{H}\to \psi\text{(2S)}~\psi \text{(2S)} are obtained via the inclusive decay of $\psi\text{(2S)}$ into $\mathrm{J/}\psi$. Furthermore, upper limits on the branching fraction for the Higgs boson decay into $\Upsilon\mathrm{(1S)}$ and $\Upsilon\mathrm{(nS)}$ ($\mathrm{n = 1, 2, 3}$) pairs are found, where for the latter all pairings of the $\Upsilon\mathrm{(1S)}$, $\Upsilon\mathrm{(2S)}$, and $\Upsilon\mathrm{(3S)}$ resonances decaying into muon pairs and the feed-down decays into lower $\Upsilon$ states are included. The decay of the $\mathrm{Z}$ boson into $\mathrm{J/}\psi$ or $\Upsilon$ pairs is also searched for.

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