A search has been performed for heavy resonances decaying to ZZ or ZW and axion-like particles mediating nonresonant ZZ or ZH production in semileptonic final states, with two charged leptons (electrons or muons) produced by the decay of a Z boson, and two quarks produced by the decay of a Z, W, or Higgs boson H. The analysis is sensitive to resonances with masses in the range from 450 to 1800 GeV. Two categories are defined based on the merged or resolved reconstruction of the hadronically decaying boson. The search is based on data collected during 2016 to 2018 by the CMS experiment at the LHC in proton-proton collisions at a center-of-mass energy of 13 TeV, corresponding to an integrated luminosity of 137 $\text{fb}^{-1}$. No significant excess is observed in the data above the standard model background expectation. Upper limits on the production cross section of heavy, narrow spin-1 and spin-2 resonances are derived as functions of the resonance mass, and exclusion limits on the production of W' bosons and bulk graviton particles are calculated in the framework of the heavy vector triplet and warped extra dimensions model, respectively. In addition, upper limits on the ALP-mediated diboson production cross section and ALP couplings to standard model particles are obtained in the framework of linear and chiral effective field theories.