Unmanned aerial vehicle (UAV) relaying network systems have been widely studied to expand network coverage and improve frequency efficiency for wireless communication. This paper proposes a carrier sense multiple access with collision avoidance (CSMA/CA) based group contention (GC) full-duplex (FD) UAV relay system. The proposed FD UAV relay system guarantees the establishment of an FD pair. The UAV acts as a relay node and notifies the next-hop destination to all nodes with the acknowledge (ACK) packet. Only nodes that can establish the FD pair with the next-hop destination compete for channel access. No matter which node wins the channel contention, throughput and delay are improved because it can guarantee the FD pair establishment. Both analytical and simulation results verified that the proposed FD UAV relay system could achieve better throughput and delay than a conventional UAV relay system.