The decays of the 125~\mathrm{GeV} Higgs boson into a Z boson and a $\rho^0(770)$ or $\phi(1020)$ meson are searched for using proton-proton collision data collected by the CMS experiment at the LHC from 2016 to 2018 at \sqrt{s}=13~\mathrm{TeV}. The analysed data set corresponds to an integrated luminosity of 137~\mathrm{fb}^{-1}. Events are selected in which the Z boson decays into a pair of electrons or a pair of muons, and the $\rho$ and $\phi$ mesons decay into pairs of pions and kaons, respectively. No significant excess above the background model is observed. As different polarization states are possible for the decay products of the Z boson and $\rho$ or $\phi$ mesons, affecting the signal acceptance, scenarios in which the decays are longitudinally or transversely polarized are considered. Upper limits at the $95\%$ confidence level on the Higgs boson branching fractions into Z$\rho$ and Z$\phi$ are determined to be $1.21-1.89\%$ and $0.36-0.58\%$, respectively, where the ranges reflect the considered polarization scenarios; these values are $860-1350$ and $860-1380$ times larger than the respective standard model expectations. These results constitute the first experimental limits on these two decay channels.

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