We consider the Peccei-Quinn (PQ) mechanism as the one behind the Dirac neutrino masses when these are generated through the $d=5$ effective operator $\overline{L}\tilde{H}{N}_R\varphi$ at one-loop level, with $\varphi$ being a Standard Model singlet scalar. In this setup, the PQ symmetry guarantees that the one-loop realization of such an effective operator gives the leading contribution to the Dirac neutrino masses by forbidding the contributions arising from its tree-level realizations. All the mediators in the one-loop neutrino mass diagrams can be stabilized by a remnant $Z_N$ symmetry from the PQ symmetry breaking, thus forming a dark sector besides the axion sector and leading to mixed axion-WIMP dark matter scenarios.