Ultrasound computed tomography has great application potential in musculoskeletal imaging owing to its advantages of low cost, real-time capability, and portability. However, the pronounced acoustic impedance contrast between soft tissue and bone in musculoskeletal structures gives rise to strong reflections, multiple scattering, and significant attenuation, resulting in a highly complex ultrasound wavefield and thereby limiting the spatial resolution and quantitative accuracy of conventional ultrasound imaging methods. To address the above issue, a quantitative ultrasound imaging method integrating contrast source inversion and full waveform inversion is proposed. This method first introduces multiplicative-regularized contrast source inversion to rapidly reconstruct the large-scale sound speed distribution of tissues under strong scattering conditions, yielding a physically consistent and robust initial model; subsequently, frequency-domain dual-parameter full waveform inversion is performed on a fine grid to achieve joint high-resolution reconstruction of sound speed and attenuation parameters, with computational efficiency enhanced through GPU-based parallel acceleration. Numerical simulations and ex vivo bovine musculoskeletal experiments demonstrate that, with a spatial grid spacing of 0.3 mm, the proposed method can effectively suppress nonlinear effects and imaging artifacts induced by high-impedance interfaces, enabling refined quantitative imaging of sound speed and attenuation distributions in musculoskeletal tissues, with a total inversion time of 16.53 min. The structural similarity indices of the reconstructed sound speed and attenuation images reach 0.954 0 and 0.902 3, respectively. The results indicate that the proposed method achieves a favorable balance between imaging accuracy and computational efficiency, providing an efficient and robust technical solution for high-resolution quantitative ultrasound imaging of musculoskeletal tissues.