  QCD running coupling constant $\alpha_s(m_c)$ and $\alpha_s(m_b)$ are
determined from heavy quarkonia $c\overline{c}$ and $b\overline{b}$ decays. The
decay rates of $V\rightarrow 3g$ and $V\rightarrow e^+ e^-$ for $V=J/\psi$ and
$\Upsilon$ are estimated by taking into account both relativistic and QCD
radiative corrections. The decay amplitudes are derived in the Bethe-Salpeter
formalism, and the decay rates are estimated by using the meson wavefunctions
which are obtained with a QCD-inspired inter-quark potential. For the
$V\rightarrow 3g$ decay we find the relativistic correction to be very large
and to severely suppress the decay rate. Using the experimental values of ratio
$R_g\equiv \frac {\Gamma (V\longrightarrow 3g)}% {\Gamma (V\longrightarrow
e^{+}e^{-})}\approx 10,~32 $ for $V=J/\psi, ~\Upsilon$ respectively, and the
calculated widths , we find $\alpha_{s}(m_c)=0.29\pm 0.02 $ and
$\alpha_s(m_b)=0.20\pm 0.02$. These values for the QCD running coupling
constant are substantially enhanced, as compared with the ones obtained without
relativistic corrections, and are consistent with the QCD scale parameter
$\Lambda_{\overline {MS}}^{(4)}% \approx 200MeV$. We also find that these
results are mainly due to kinematic corrections and not sensitive to the
dynamical models.
