Next-generation networks need to meet ubiquitous and high data-rate demand. Therefore, this paper considers the throughput and trajectory optimization of terahertz (THz)-enabled unmanned aerial vehicles (UAVs) in the sixth-generation (6G) communication networks. In the considered scenario, multiple UAVs must provide on-demand terabits per second (TB/s) services to an urban area along with existing terrestrial networks. However, THz-empowered UAVs pose some new constraints, e.g., dynamic THz-channel conditions for ground users (GUs) association and UAV trajectory optimization to fulfill GU's throughput demands. Thus, a framework is proposed to address these challenges, where a joint UAVs-GUs association, transmit power, and the trajectory optimization problem is studied. The formulated problem is mixed-integer non-linear programming (MINLP), which is NP-hard to solve. Consequently, an iterative algorithm is proposed to solve three sub-problems iteratively, i.e., UAVs-GUs association, transmit power, and trajectory optimization. Simulation results demonstrate that the proposed algorithm increased the throughput by up to 10%, 68.9%, and 69.1% respectively compared to baseline algorithms.