Utilizing unmanned aerial vehicles (UAVs) as mobile access points or base stations has emerged as a promising solution to address the excessive traffic demands in wireless networks. This paper investigates improving the detector performance at the unmanned aerial vehicle base stations (UAV-BSs) in an uplink grant-free non-orthogonal multiple access (GF-NOMA) system by considering the activity state (AS) temporal correlation of the different user equipments (UEs) in the time domain. The Bernoulli Gaussian-Markov chain (BG-MC) probability model is used for exploiting both the sparsity and slow change characteristic of the AS of the UE. The GAMP Bernoulli Gaussian-Markov chain (GAMP-BG-MC) algorithm is proposed to improve the detector performance, which can utilize the bidirectional message passing between the neighboring time slots to fully exploit the temporally correlated AS of the UE. Furthermore, the parameters of the BG-MC model can be updated adaptively during the estimation procedure with unknown system statistics. Simulation results show that the proposed algorithm can improve the detection accuracy compared to existing methods while keeping the same order complexity.