  We study the $\mu$-$e$ conversion in muonic atoms via an exchange of family
gauge boson (FGB) $A_{2}^{\ 1}$ in a $U(3)$ FGB model. Within the class of FGB
model, we consider three types of family-number assignments for quarks. We
evaluate the $\mu$-$e$ conversion rate for various target nuclei, and find that
next generation $\mu$-$e$ conversion search experiments can cover entire energy
scale of the model for all of types of the quark family-number assignments. We
show that the conversion rate in the model is so sensitive to up- and
down-quark mixing matrices, $U^{u}$ and $U^{d}$, where the CKM matrix is given
by $V_\text{CKM} = U^{u\dagger} U^d$. Precise measurements of conversion rates
for various target nuclei can identify not only the types of quark
family-number assignments, but also each quark mixing matrix individually.
