= Solution
A <continuous-time Markov chain> is a <reversible Markov chain> in equilibrium if, when started in a <stationary distribution> $\pi$, the process $(X_t:0\leq t\leq T)$ has the same finite-dimensional distributions as $(X_{T-t}:0\leq t\leq T)$ for every $T>0$.
For transition rates $q_{ij}$, the <detailed balance for a continuous-time Markov chain> equations are
$$
\boxed{\pi_iq_{ij}=\pi_jq_{ji}\qquad(i\ne j).}
$$
They imply invariance directly. For each state $j$,
$$
\begin{aligned}
(\pi Q)_j
&=\sum_{i\ne j}\pi_iq_{ij}+\pi_jq_{jj}\\
&=\pi_j\sum_{i\ne j}q_{ji}
-\pi_j\sum_{k\ne j}q_{jk}=0,
\end{aligned}
$$
because $q_{jj}=-\sum_{k\ne j}q_{jk}$. Thus $\pi Q=0$, so $\pi$ is an <invariant distribution of a continuous-time Markov chain>.
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