  In this paper, we study the non-leptonic $B_{c}\to J/\psi V$ and
$B_{c}^{*}\to\eta_{c}V$~$(V=\rho\,,K^{*})$ weak decays in the framework of QCD
factorization. In the evaluation, the form factors are calculated by using the
Bauer-Stech-Wirbel model and the light-front quark model, respectively. Besides
the longitudinal amplitude, the power-suppressed transverse contributions are
also evaluated at next-to-leading order. The predictions for the observables of
$B_{c}\to J/\psi V$ and $B_{c}^{*}\to\eta_{c}V$ decays are presented. We find
that the NLO QCD contribution presents about $ 8\%$ correction to the branching
ratios, and the longitudinal polarization fractions of these decays are at the
level of $(80\sim90)\%$. In addition, we suggest direct measurements on some
useful ratios, $R_{K^{*}/\rho}^{(\lambda=0)}$ and
$\widetilde{R}^{(\lambda=0)}_{K^{*}/\rho}$, which are very suitable for test
the consistence between theoretical prediction and data because their
theoretical uncertainties can be well-controlled.
