  We report $^{121,123}$Sb nuclear quadrupole resonance (NQR) measurements in
the filled-skutterudite superconductor PrOs$_4$Sb$_{12}$ in the temperature
range of 0.05-30 K. The electric field gradients (EFG), $V_{zz}$ and
$V_{xx}-V_{yy}$, at the Sb site exhibit unusual temperature dependence below 30
K. To explain these features, we discuss the coupling between the Sb nuclear
quadrupole moment and Pr $4f^2$-derived multipole moments. The observed $T$
dependence of EFG is well explained by the CEF quasi-quartet consisted of
$\Gamma_1$ singlet and $\Gamma_4^{(2)}$ triplet states in the cubic point group
$T_h$. These results, in turn, are indicative of the importance of the coupling
between the $^{121,123}$Sb quadrupole moments and the hexadecapole moment
caused by the quasi-quartet.
