This article studies an efficient nonlinear model-predictive control (NMPC) scheme for trajectory tracking control of a quadrotor unmanned aerial vehicle (UAV). By augmenting the desired trajectory to a reference dynamical system, we can make the tracking task fit into the standard NMPC framework. In order to alleviate the heavy computational burden caused by solving the corresponding NMPC optimization problem online, we develop an improved continuation/generalized minimal residual (<inline-formula> <tex-math notation="LaTeX">{i}\text{C} </tex-math></inline-formula>/GMRES) algorithm. Compared with the standard C/GMRES method, the inequality constraint is relaxed by imposing the penalty term on the cost function. To guarantee the closed-loop system stability, we introduce a contraction constraint. Based on the proposed numerical algorithm and the stability constraint, we develop a novel efficient-NMPC algorithm to achieve acceptable control performance with reduced computational complexity. The numerical convergence of <inline-formula> <tex-math notation="LaTeX">{i}\text{C} </tex-math></inline-formula>/GMRES solutions and the closed-loop stability of efficient-NMPC are theoretically analyzed in the presence of the input constraint. Finally, the numerical simulations, software-in-the-loop (SIL) simulations, and the real-time experiment are given to demonstrate the effectiveness of the proposed <inline-formula> <tex-math notation="LaTeX">{i}\text{C} </tex-math></inline-formula>/GMRES algorithm and efficient-NMPC scheme.