In modern industrial and engineering systems, stochastic degradation models are widely used for reliability analysis and maintenance decision-making. However, due to imperfect sensors and environmental influences, it is difficult to directly observe the latent degradation states. Traditional degradation models typically assume that measurement errors have simple statistical properties, but this assumption often does not hold in practical applications. To address this issue, this paper constructs a degradation model based on the Gamma process (GP) and assumes that measurement noise can be characterized by the fractional Gaussian noise (FGN). Furthermore, this paper proposes a method combining Gibbs sampling with the stochastic expectation-maximization (SEM) algorithm to achieve efficient estimation of the model parameters and accurate inference of the latent degradation states. Simulation results demonstrate that the proposed model, validated solely through numerical simulations, exhibits improved generalizability compared to the GP model with Gaussian noise.