This letter proposes a new full-duplex (FD) secrecy communication scheme for the unmanned aerial vehicle (UAV) and investigates its optimal design to achieve the maximum energy efficiency (EE) of the UAV. Specifically, the UAV receives the confidential information from a ground source and meanwhile sends jamming signals to interfere with a potential ground eavesdropper. As the UAV has limited on-board energy in practice, we aim to maximize the EE for its secrecy communication, by jointly optimizing the UAV trajectory and the source/UAV transmit/jamming powers over a finite flight period with given initial and final locations. Although the problem is difficult to solve, we propose an efficient iterative algorithm to obtain its suboptimal solution. Simulation results show that the proposed joint design can significantly improve the EE of UAV secrecy communication, as compared to various benchmark schemes.