This work introduces a framework for analyzing the Age of Incorrect Information (AoII) in a real-time monitoring system with a generic discrete-time Markov source. We study a noisy communication system employing a hybrid automatic repeat request (HARQ) protocol, subject to a transmission rate constraint. The optimization problem is formulated as a constrained Markov decision process (CMDP), and it is shown that there exists an optimal policy that is a randomized mixture of two stationary policies. To overcome the intractability of computing the optimal stationary policies, we develop a multiple-threshold policy class where thresholds depend on the source, the receiver, and the packet count. By establishing a Markov renewal structure induced by threshold policies, we derive closed-form expressions for the long-term average AoII and transmission rate. The proposed policy is constructed via a relative value iteration algorithm that leverages the threshold structure to skip computations, combined with a bisection search to satisfy the rate constraint. To accommodate scenarios requiring lower computational complexity, we adapt the same technique to produce a simpler single-threshold policy that trades optimality for efficiency. Numerical experiments exhibit that both thresholdbased policies outperform periodic scheduling, with the multiplethreshold approach matching the performance of the globally optimal policy.