  A scotogenic model can radiatively generate the observed neutrino mass,
provide a dark matter candidate, and lead to rare lepton flavor-violating
processes. We aim to extend the model to establish a potential connection to
the quark flavor-related processes within the framework of scotogenesis,
enhancing the unexpectedly large branching ratio (BR) of $B^+\to K^+ \nu
\bar\nu$, observed by Belle II Collaboration. Meanwhile, the model can address
tensions between some experimental measurements and standard model (SM)
predictions in flavor physics, such as the muon $g-2$ excess and the higher BR
of $B_s \to \mu^- \mu^+$. We introduce in the model the following dark
particles: a neutral singlet Dirac-type lepton ($N$); two inert Higgs doublets
($\eta_{1,2}$), with one of which carrying a lepton number; a charged singlet
dark scalar $(\chi^+)$, and a singlet vector-like up-type dark quark ($T$). The
first two entities are responsible for the radiative neutrino mass, and
$\chi^+$ couples to right-handed quarks and leptons and can resolve the
tensions existing in muon $g-2$ and $B_s\to \mu^- \mu^+$. Furthermore, the BR
of $B^+ \to K^+ \nu \bar\nu$ can be enhanced up to a factor of 2 compared to
the SM prediction through the mediations of the dark $T$ and the charged
scalars. In addition, we also study the impacts on the $K\to \pi \nu \bar\nu$
decays.
