We report experimental results of nuclear magnetic resonance (NMR) at the La site and nuclear quadrupole resonance (NQR) at the Sb site in the filled skutterudite LaOs$_4$Sb$_{12}$. We found that the nuclear spin-lattice relaxation rate divided by temperature $1/T_1T$ at the La site exhibits a different temperature dependence from that at the Sb site. Although $1/T_1T$ at the Sb site is explained by the Korringa mechanism, $1/T_1T$ at the La site exhibits a broad maximum around 50 K, showing the presence of an additional contribution at the La site. The additional low-lying excitations observed at the La site can be understood with the relaxation from anharmonic phonons due to the rattling motion of the La atoms.