Consider a continuous-time Markov chain (CTMC) $X$ on a countably infinite state space $S$. The CTMC is irreducible and all the states are positive recurrent. Let $T(i,j)$ be the first passage time to state $j$ given that it starts in state $i$. $T(i,i)$ should be understood to be the return time to state $i$.
Since all states are positive recurrent, we have for all $i \in S$,
\begin{equation} \mathbb{E}[T(i,i)] < \infty \quad \Leftrightarrow \quad \mathbb{P}(T(i,i) < \infty) = 1. \end{equation}
Intuitively, it seems that we should also have for all $i,j \in S$,
\begin{equation} \mathbb{E}[T(i,j)] < \infty \quad \Leftrightarrow \quad \mathbb{P}(T(i,j) < \infty) = 1. \end{equation}
However, I don't see an immediate way to prove this. Possibly by using that the $\mathbb{P}(X(t) = j \mid X(0) = i) > 0$ for all $i,j \in S$ and $t > 0$ by positive recurrence?