For the quadrotor unmanned aerial vehicle (UAV) system with time-varying load, the finite-time control strategy is studied in the presence of unknown external disturbances and unknown drag coefficients. Firstly, a complete mathematical model of quadrotor UAV through the Newton-Euler method is established. An adaptive trajectory tracking controller with unknown drag coefficient is designed by combining the adaptive parameter correction method of position loop with backstepping control to estimate load. Then, in order to improve the convergence rate and reduce the impact of the external disturbances, a finite-time sliding mode controller is proposed based on the disturbance observers. By using the Lyapunov stability theory, the position loop and the attitude loop are proved to be asymptotically stable and finite-time stable, respectively. Finally, the effectiveness and superiority of the proposed controller are validated by numerical simulations.