A search for the production of a narrow-width resonance decaying into a pair of Higgs bosons decaying into the (bbZZ) channel is presented. The analysis is based on data collected with the CMS detector during 2016, in proton-proton collisions at the LHC, corresponding to an integrated luminosity of (35.9\text{}\text{}{\mathrm{fb}}^{-1}). The final states considered are the ones where one of the (Z) bosons decays into a pair of muons or electrons, and the other (Z) boson decays to either a pair of quarks or a pair of neutrinos. Upper limits at 95% confidence level are placed on the production of narrow-width spin-0 or spin-2 particles decaying to a pair of Higgs bosons, in models with and without an extended Higgs sector. For a resonance mass range between 260 and 1000 GeV, limits on the production cross section times branching fraction of a spin-0 and spin-2 resonance range from 0.1 to 5.0 pb and 0.1 to 3.6 pb, respectively. These results set limits in parameter space in bulk Randall-Sundrum radion, Kaluza-Klein excitation of the graviton, and next-to-minimal two-Higgs doublet models (N2HDMs). For specific choices of parameters the N2HDM can be excluded in a mass range between 360 and 620 GeV for a resonance decaying to two Higgs bosons. This is the first search for Higgs boson resonant pair production in the (bbZZ) channel.