  We analyze the strong CP problem and the implications for axion physics in
the context of $U_1$ vector leptoquark models, recently put forward as an
elegant solution to the hints of lepton flavor universality violation in
B-meson decays. It is shown that in minimal gauge models containing the $U_1$
as a gauge boson, the Peccei-Quinn solution of the strong CP problem requires
the introduction of two axions. Characteristic predictions for the associated
axions can be deduced from the model parameter space hinted by B-physics,
allowing the new axion sector to account for the dark matter of the Universe.
We also provide a specific ultraviolet completion of the axion sector that
connects the Peccei-Quinn mechanism to the generation of neutrino masses.
