In this paper, we study the resource allocation and trajectory design for energy-efficient secure unmanned aerial vehicle (UAV) communication systems where a UAV base station serves multiple legitimate ground users in the existence of a potential eavesdropper. We aim to maximize the energy efficiency of the UAV by jointly optimizing its transmit power, user scheduling, trajectory, and velocity. The design is formulated as a non-convex optimization problem taking into account the maximum tolerable signal-to-noise ratio (SNR) leakage, the minimum data rate requirement of each user, and the location uncertainty of the eavesdropper. An iterative algorithm is proposed to obtain an efficient suboptimal solution. Simulation results demonstrate that the proposed algorithm can achieve a significant improvement of the system energy efficiency while satisfying communication security constraint, compared to some simple scheme adopting straight flight trajectory with a constant speed.