Euler and Cardan angles representation in biomechanical analysis allows straightforward description of joint rotations. However, application of Euler or Cardan angles can be problematic due to a singularity called gimbal lock. Quaternions offer an alternative way to describe rotation that avoids this problem, but they are not commonly used in biomechanics as they are complex and not inherently intuitive, specifically in dynamic models actuated by muscles. This study introduces a mathematical framework for describing muscle actions in dynamic quaternion-based musculoskeletal simulations. The proposed method estimates muscle torques in a quaternion-based musculoskeletal model. Its application is shown in a three-dimensional double-pendulum system actuated by muscle elements. Furthermore, the transformation of muscle moment arms obtained from muscle paths based on Euler or Cardan angles into a quaternion-based description is presented. The proposed method is advantageous for dynamic modeling of musculoskeletal models with complex kinematics and joints with large ranges of motion like the shoulder joint.