The charged axial-vector J (P) = 1(+) tetraquarks Z(q) = [cq][ (b) over bar(q) over bar] and Z(s) = [cs][ (b) over bar(s) over bar] 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 aQCDtwo-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 gZ(q) B-c rho and gZ(s) B-c phi couplings. For these purposes, the QCD light-cone sum rule method and its soft-meson approximation are utilized. The couplings gZ(q) B-c rho and gZ(s) B-c phi, extracted from this analysis, are applied for evaluating the strong Z(q) -> B-c rho and Z(s) -> 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.