  In this article, we study momentum dependence of the hadronic coupling
constants $G_{B_c^* B_c \Upsilon}$, $G_{B_c^* B_c J/\psi}$, $G_{B_c B_c
\Upsilon}$ and $G_{B_c B_c J/\psi}$ with the off-shell $\Upsilon$ and $J/\psi$
using the three-point QCD sum rules. Then we fit the hadronic coupling
constants into analytical functions and extrapolate them into deep time-like
regions to obtain the on-shell values
  $ G_{B_c^*B_c\Upsilon}(q^2=M_{\Upsilon}^2)$, $ G_{B_c^*B_c
J/\psi}(q^2=M_{J/\psi}^2)$, $ G_{B_cB_c\Upsilon}(q^2=M_{\Upsilon}^2)$ and $
G_{B_cB_c J/\psi}(q^2=M_{J/\psi}^2)$ for the first time. Those hadronic
coupling constants can be taken as basic input parameters in phenomenological
analysis.
