Decays of the 125 GeV Higgs boson into a Z boson and a ρ0(770) or ϕ(1020) meson are searched for using proton-proton collision data collected by the CMS experiment at the LHC at s = 13 TeV. The analysed data set corresponds to an integrated luminosity of 137 fb−1. Events are selected in which the Z boson decays into a pair of electrons or a pair of muons, and the ρ and ϕ 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 ρ or ϕ 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 Zρ and Zϕ are determined to be 1.04–1.31% and 0.31–0.40%, respectively, where the ranges reflect the considered polarization scenarios; these values are 740–940 and 730–950 times larger than the respective standard model expectations. These results constitute the first experimental limits on the two decay channels. [Figure not available: see fulltext.]