Previous front-tracking (FT) method-based models to simulate droplet motion on a solid surface with a moving contact line (MCL) are limited to two-dimensional models in which the Navier boundary condition (NBC) is employed for the MCL. In this paper, we develop a three-dimensional FT method that integrates the generalized Navier boundary condition (GNBC) to model the MCL. This GNBC-based FT method addresses several key issues, such as the integration of GNBC for the dynamic description of the MCL and its coupling with the surrounding flow, the accurate updating of the density and viscosity of the two-phase fluid near the contact line, and the restructuring of the Lagrangian mesh for tracking the drop surface, especially near the contact line. The stability and accuracy of the present numerical method are validated by several tests: (1) numerical performance tests, (2) simulation of the transient and steady-state shapes of droplets under flow with a fixed contact line, and (3) simulation of a droplet spreading under gravity and moving under a shear flow with MCLs. Excellent agreement is achieved between the results obtained by our model and the data obtained by other theoretical and numerical approaches.