  In the framework of Poincar\'{e} covariant quark model the behavior of
running coupling constant $\alpha_{s}\left(Q^{2}\right)$ is considered in $Q <
1 \mbox{GeV}$ region. An analysis was carried out for pseudoscalar and vector
mesons with lepton decay constants, masses (obtained from model dependent) and
nucleon spin rules required to match their experimental counterparts. Possible
behavior of $\alpha_{s}$ with
$\alpha_{\mathrm{crit.}}=\alpha_{s}\left(Q^{2}=0\right) \sim 0.65-0.72$ in the
case of a frozen regime, which follows from experimental values of lepton decay
constant, masses and nucleon spin rules are discussed.
