We fabricated an acousto-optic semiconductor hybrid device for strong optomechanical coupling of individual quantum emitters and a surface acoustic wave. Our device comprises a surface acoustic wave chip made from highly piezoelectric LiNbO$_3$ and a GaAs-based semiconductor membrane with an embedded layer of quantum dots. Employing multi-harmonic transducers, we generated sound waves on LiNbO$_3$ over a wide range of radio frequencies. We monitored their coupling to and propagation across the semiconductor membrane both in the electrical and optical domain. We demonstrate enhanced optomechanical tuning of the embedded quantum dots with increasing frequencies. This effect was verified by finite element modelling of our device geometry and attributed to an increased localization of the acoustic field within the semiconductor membrane. For moderately high acoustic frequencies, our simulations predict strong optomechanical coupling making our hybrid device ideally suited for applications in semiconductor based quantum acoustics.