The sixth generation (6G) communication networks are expected to provide high data rates, ultra-reliable communication, and massive connectivity, especially in challenging environments such as dense urban areas and disaster-affected regions. However, traditional terrestrial-only networks face significant challenges in these scenarios, including signal blockages from high-rise buildings, traffic congestion, and dynamic user distributions. To address these limitations, we propose the adaptive multi-UAV deployment (AMUD) framework within satellite air-ground integrated networks (SAGINs). The AMUD framework dynamically deploys amplify-and-forward multiple unmanned aerial vehicle relay (UAVr) in with low Earth orbit (LEO) satellites to improve coverage, alleviate congestion, and ensure reliable communication in non-line-of-sight and high-demand conditions. We formulate an optimization problem that aims to jointly maximize the energy efficiency of the total network and the total capacity while ensuring the fairness of the total capacity and satisfying the users' requirements. The simulation results demonstrate that AMUD improves the total capacity of the network, improves the total energy efficiency, and increases the fairness of the capacity compared to traditional LEO satellite and ground base station (LEO-GBS) only systems.