We report $^{123}$Sb nuclear quadrupole resonance (NQR) measurements of the filled skutterudite heavy-fermion superconductor PrOs$_4$Sb$_{12}$ under high pressure. The temperature dependence of NQR frequency and the spin-lattice relaxation rate $1/T_1$ indicate that the crystal-electric-field splitting $Delta_{rm CEF}$ between the ground state $Gamma_1$ singlet and the first excited state $Gamma_4^{(2)}$ triplet decreases with increasing pressure. Ac-susceptibility measurements indicate that the superconducting transition temperature ($T_{rm c}$) also decreases with increasing pressure. However, above $P$ $sim$ 2 GPa, both $Delta_{rm CEF}$ and $T_{rm c}$ do not depend on external pressure up to $P$ = 3.82 GPa. These pressure dependences of $Delta_{rm CEF}$ and $T_{rm c}$ suggest an intimate relationship between quadrupole excitations associated with the $Gamma_4^{(2)}$ level and unconventional superconductivity in PrOs$_4$Sb$_{12}$. In the superconducting state, 1/$T_1$ below $T_{rm c}$ = 1.55 and 1.57 K at $P$ = 1.91 and 2.63 GPa shows a power-law temperature variations and are proportional to $T^5$ at temperatures considerably below $T_{rm c}$. These data can be well fitted by the gap model $Delta(theta) = Delta_0sintheta$ with $Delta_0$ = 3.08 $k_{rm B}T_{rm c}$ and 3.04 $k_{rm B}T_{rm c}$ for $P$ = 1.91 and 2.63 GPa, respectively. The results indicate there exists point nodes in the gap function.
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