We propose a model of Dirac neutrino masses generated at one-loop level. The origin of this mass is induced from Peccei-Quinn symmetry breaking which was proposed to solve the so-called strong CP problem in QCD; therefore, the neutrino mass is connected with the QCD scale, ΛQCD. We also study the parameter space of this model, confronting neutrino oscillation data and leptonic rare decays. The phenomenological implications to leptonic flavor physics, such as the electromagnetic moment of charged leptons and neutrinos, are studied. Axion as the dark matter candidate is one of the byproducts in our scenario. Diphoton and Z-photon decay channels in the LHC Higgs search are investigated. We show that the effects of a singly charged singlet scalar can be distinguished from the general two Higgs doublet model.