The NMR spectra of $Li_3Rb(SO_4)_2$ crystals and their relaxation processes were investigated by using $^7Li$ and $^{87}Rb$ NMR. The relaxation rates of the $^7Li$ and $^{87}Rb$ nuclei in the crystals were found to increase with increasing temperature, and can be described by the relation $T_1^{-1}{propto}AT^2$. The dominant relaxation mechanism for these nuclei with electric quadrupole moments is provided by the coupling of these moments to the thermal fluctuations of the local electric field gradient via Raman spin-phonon processes.