There are two types of events labelled $A$ and $B$. When an event of type $A$ happens there is a higher chance the next event will also be $A$ given by the transition matrix $p(A\to B)$ and $p(B \to A)$. Now, there are three counting processes $N_A$, $N_B$ and $N$, counting the number of events labeled $A$, $B$ and all events correspondingly. Events (irregardless of their label) come independently so that $N$ is a counting process of a homogenuous Poisson process.
How to describe the processes of $N_A$ and $N_B$? What are their intensities?
Since I still don’t fully understand what you want to calculate after our exchange in the comments, I’ll just calculate what I can and you can tell me whether it includes what you wanted :-)
The marginal (i.e. unconditional) intensity of the $A$ process is $p_A N$, where $p_A$ is the equilibrium distribution of the Markov process with the given transition matrix between states $A$ and $B$.
The intensity of the $A$ process conditional on the last event having been an $A$ event is $p(A\to A)N$.
The interesting bit is the intensity of the $A$ process conditional on there having been an $A$ event at time $\tau$. (The difference from the previous case is that this wasn’t necessarily the last event; there might have been intervening events of either type.)
Define the state of the process to be the last type of event that occurred. (Here I’m introducing an asymmetry in time by looking at the last instead of the next event, but that’s just because I find it easier to think about these things that way – the conditional probabilities are symmetric in the time difference, i.e. the conditional probability of an $A$ event at $\tau+\delta t$ is the same as at $\tau-\delta t$.)
This state is subject to a continuous-time Markov process with transition rate $\alpha=Np(A\to B)$ from state $A$ to state $B$ and $\beta=NP(B\to A)$ from state $B$ to state $A$. As explained in the Wikipedia article, the resulting occupancy in state $A$ is
$$\frac{\beta}{\alpha+\beta} + \frac{\alpha}{\alpha+\beta}e^{-(\alpha+\beta)(t-\tau)}\;,$$
an exponential decay from occupancy $1$ immediately after the event being conditioned on to the equilibrium occupancy of the discrete Markov process. The intensity of the $A$ process conditional on an $A$ event at time $\tau$ is $N$ times this occupancy.