Passively Q-switched self-Raman laser oscillations in the SrMoO4:Nd3+ crystal optically pumped by a laser diode (LD) at 804 nm are demonstrated for the first time to our knowledge. Output parameters of LD-pumped SrMoO4:Nd3+ laser were investigated in free-running, passively Q-switched, and self-Raman regimes of laser operation, over a wide range of pulse repetition rates (8.5-85 kHz). A pulse energy of 21 microJ was measured in the self-Raman regime at 1163 nm (first Stokes) wavelength, which is 6 times higher than that for the GdVO4:Nd3+ self-Raman laser of similar design.