Decays of the Higgs boson into a [Formula presented] boson and a [Formula presented] or [Formula presented] meson are searched for in four-lepton final states with the CMS detector at the LHC. A data set of proton-proton collisions corresponding to an integrated luminosity of 138fb−1 is used. Using the same data set, decays of the Higgs and [Formula presented] boson into quarkonium pairs are also searched for. An observation of such decays with this sample would indicate the presence of physics beyond the standard model. No evidence for these decays has been observed and upper limits at the 95% confidence level are placed on the corresponding branching fractions (B). Assuming longitudinal polarization of the Higgs boson decay products, 95% confidence level observed upper limits for [Formula presented] and [Formula presented] are 1.9×10−3 and 6.6×10−3, respectively.