  The axial-vector mesons $f_1(1285)$ and $f_1(1420)$ are particularly viewed
as the mixtures of flavor states $f_n$ and $f_s$ with mixing angle $\varphi$.
In order to determine this angle, we study the $B_{d,s}^0\to J/\psi
f_1(1285,1420)$ and $B_{d,s}^0\to \eta_c f_1(1285,1420)$ decays in perturbative
QCD (PQCD) approach, including the effects of vertex corrections,
nonfactorizable diagrams and penguin operators. Not only the branching
fractions, but also the direct $CP$ asymmetries and the polarization fractions
are calculated. It is found that the branching fractions of these decays are
large enough to be measured in the running LHCb and Belle-II experiments.
Moreover, in comparison with the observed ${\cal B}(B_{d,s}^0 \to J/\psi
f_1(1285))$, $B_s^0 \to (J/\psi, \eta_c) f_1(1420)$ decays have large branching
fractions, which could be measured promisingly through $f_1(1420) \to K_S^0
K^\pm \pi^\mp$ in experiments. We also propose several ratios that could be
used to further constrain the absolute value of the mixing angle $\varphi$, but
its sign cannot be determined yet in these decays. The direct CP asymmetries of
these decays indicate the penguin pollution in the $B_d^0 \to (J/\psi, \eta_c)
f_1$ decays cannot be neglected. We acknowledge that there are large
theoretical uncertainties arising from the distribution amplitudes of
axial-vector mesons and charmonium states, and more precise nonperturbative
parameters are called. The comparisons between our results and future
experimental data would help us to understand the nature of $f_1$ states and to
test the PQCD approach.
