An efficient monolithic Q-switched Nd : MgO : LiNbO3 waveguide laser operating at 1.085 mum 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 mum coupled pump power. This source was frequency-doubled in an external quasiphase-matched LiNbO3 waveguide yielding 12 W pulses at 0.54 mum with a 15% conversion efficiency.