An efficient monolithic Q-switched Nd:MgO:LiNbO3 waveguide laser operating at 1.085 μm is reported. Q-switched pulses of 350 W peak power and 300 ps width are obtained for less than 15 mW of CW 0.814 μm coupled pump power. This source was frequency-doubled in an external quasiphase-matched LiNbO3 waveguide yielding 12 W pulses at 0.54 μm with a 15% conversion efficiency.