This note considers the problem of evaluating a probabilistic distance between discrete-time, homogeneous, first-order, finite-state finite-alphabet hidden Markov models (HMMs). Our approach is based on a correspondence between probability measures and HMMs established in this note. Using a probability measure transformation technique, we obtain recursive expressions for the relative entropy between the marginal probability distributions of two HMMs under consideration. Also, the relative entropy rate, as the limit of the time-averaged value of the above relative entropy, is obtained. These expressions are given in terms of the parameters of the given HMMs. Furthermore, we show that the probabilistic distance between HMMs used in the existing literature admits a representation in terms of a conditional expectation given the observation sequence. This representation allows us to evaluate this distance using an information state approach.