A search for new physics in events with two highly Lorentz-boosted Z bosons and large missing transverse momentum is presented. The analyzed proton-proton collision data, corresponding to an integrated luminosity of 137 fb(-1), were recorded at s = 13 TeV by the CMS experiment at the CERN LHC. The search utilizes the substructure of jets with large radius to identify quark pairs from Z boson decays. Backgrounds from standard model processes are suppressed by requirements on the jet mass and the missing transverse momentum. No significant excess in the event yield is observed beyond the number of background events expected from the standard model. For a simplified supersymmetric model in which the Z bosons arise from the decay of gluinos, an exclusion limit of 1920 GeV on the gluino mass is set at 95% confidence level. This is the first search for beyond-standard-model production of pairs of boosted Z bosons plus large missing transverse momentum.