Unmanned aerial vehicle (UAV) communication is a research hotspot in the field of wireless communication. In order to ensure the reliability of communication between ground mobile terminal and UAV, a cooperative relay transmission scheme based on space-time coding (STBC) is proposed in this paper. Then, by using cognitive radio technology to improve the spectral efficiency, we formulate an optimization problem to maximize the throughput of cognitive relay network under the constraints of the quality-of-service (QoS) requirement of the primary users and the transmission power limitation of cognitive users with amplify-and-forward (AF) and decode-and-forward (DF) protocols. The Lagrange multiplier function and Karush-Kuhn-Tucker (KKT) condition are combined to solve the optimal power allocation. Finally, the simulation results not only verify the effectiveness of the proposed scheme, but also show that the throughput of cognitive relay network with DF protocol is better than that of AF protocol.