This paper considers Distributed Hypothesis Testing (DHT), in which a source X is encoded given that a side information Y is available only to the decoder. The decoder's task is to make a decision between two hypothesis H0 and H1 related to the joint probability distribution of X and Y. While most works on DHT have adopted an information-theoretic approach by providing generic error exponents for the decision error, we focus on a specific and practical source model: the Gilbert-Elliot (GE) model. The later is a two-states hidden Markov model which describes time-varying correlation between X and Y. Starting from the generic error exponents, we develop a method for numerically estimating the error exponent for the GE model. We provide numerical results to evaluate the impact of the model parameters on the error exponents and explore the tradeoff between two types of error events, namely the testing error and the binning error.