This note presents a search for new resonances decaying into a $Z$ boson and a 125~ GeV Higgs boson in the $\nu\nu bb$ and $\ell\ell bb$ final states, where $\ell=e^{\pm}$ or $\mu^{\pm}$, in $pp$ collisions at \sqrt{s}=13~TeV. The data used correspond to a total integrated luminosity of 139~\ifb\ collected by the ATLAS detector during Run-2 of the Large Hadron Collider. The search is conducted by examining the reconstructed invariant or transverse mass distributions of $Zh$ candidates for evidence of a localised excess in the mass range from 300~{GeV} to 5~{TeV}. No significant excess is observed and $95\%$ confidence level upper limits between 1~pb and 0.4~fb are placed on the production cross-section times branching fraction of $Z^{'}$ resonances in heavy-vector-triplet models and the CP-odd scalar boson $A$ in two-Higgs-doublet models.}

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