  The charged axial-vector $J^{P}=1^{+}$ tetraquarks $Z_{q}=[cq][\bar {b} \bar
q ]$ and $Z_{s}=[cs][\bar {b} \bar s]$ with the open charm-bottom contents are
studied in the diquark-antidiquark model. The masses and meson-current
couplings of these states are calculated by employing QCD two-point sum rule
approach, where the quark, gluon and mixed condensates up to eight dimensions
are taken into account. These parameters of the tetraquark states $ Z_{q}$ and
$Z_{s}$ are used to analyze the vertices $Z_q B_c \rho $ and $Z_s B_c \phi $ to
determine the strong $g_{Z_qB_c \rho }$ and $g_{Z_sB_c \phi }$ couplings. For
these purposes, QCD light-cone sum rule method and its soft-meson approximation
are utilized. The couplings $g_{Z_qB_c \rho }$ and $ g_{Z_sB_c \phi }$,
extracted from this analysis, are applied for evaluating of the strong $Z_q \to
B_c \rho$ and $Z_s \to B_c \phi$ decays' widths, which are essential results of
the present investigation. Our predictions for the masses of the $Z_{q}$ and
$Z_{s}$ states are confronted with similar results available in the literature.
