We investigate the optomechanical properties of tensile-strained ternary InxGa1−xP nanomembranes grown on GaAs. This material system combines the benefits of highly strained membranes, similar to those based on stoichiometric silicon nitride, with the unique properties of thin-film semiconductor single crystals, as previously demonstrated with suspended GaAs. Here, we employ lattice mismatch in epitaxial growth to impart an intrinsic tensile strain to a monocrystalline thin film (approximately 30 nm thick). These structures exhibit mechanical quality factors of 2 × 106 or beyond at room temperature and 17 K for eigenfrequencies up to 1 MHz, yielding Q × f products of 2 × 1012 Hz for a tensile stress of ∼170 MPa. Incorporating such membranes in a high-finesse Fabry-Perot cavity, we extract an upper limit to the total optical loss (including both absorption and scatter) of 40 ppm at 1064 nm and room temperature. Further reductions of the In content of this alloy will enable tensile stress levels of 1 GPa, with the potential for a significant increase in the Q × f product, assuming no deterioration in the mechanical loss at this composition and strain level. This materials system is a promising candidate for the integration of strained semiconductor membrane structures with low-loss semiconductor mirrors and for realizing stacks of membranes for enhanced optomechanical coupling.