A measurement is presented of the electroweak vector boson scattering production of ZV (V = W, Z) boson pairs associated with two jets in proton-proton collisions at a center-of-mass energy of 13 TeV. The data, corresponding to an integrated luminosity of 138 fb$^{-1}$, were collected at the CERN LHC with the CMS detector during the 2016--2018 data-taking period. The analysis targets final states with a pair of isolated electrons or muons from Z boson decays and three or four jets, depending on the momentum of the vector boson that decays into quarks. Signal strength is measured for events characterized by a large invariant mass of two forward jets with a wide pseudorapidity gap between them. The electroweak production of ZV in association with two jets is measured with an observed (expected) significance of 1.3 (1.8) standard deviations. A combination of the analyses of ZV channel and the previously published WV channel in the lepton plus jets final state places constraints on effective field theory parameters that describe anomalous electroweak production of WW, WZ, and ZZ boson pairs in association with two jets. Several world best limits are set on anomalous quartic gauge couplings in terms of dimension-8 standard model effective field theory operators.