This paper investigates Distributed Hypothesis testing (DHT), in which a source X is encoded given that side information Y is available at the decoder only. Based on the received coded data, the receiver aims to decide on the two hypotheses H0 or H1 related to the joint distribution of X and Y. While most existing contributions in the literature on DHT consider i.i.d. assumptions, this paper assumes more generic, non-i.i.d., non-stationary, and non-ergodic sources models. It relies on information-spectrum tools to provide general formulas on the achievable Type-II error exponent under a constraint on the Type-I error. The achievability proof is based on a quantize-and-binning scheme. It is shown that with the quantize-and-binning approach, the error exponent boils down to a trade-off between a binning error and a decision error, as already observed for the i.i.d. sources. The last part of the paper provides error exponents for particular source models, e.g., Gaussian, stationary, and ergodic models.