Ben correctly points out that a Markov chain is Markovian both forwards and backwards (+1). This is always true, but it is not the same thing as reversibility. In particular, reversibility requires the existence of a stationary distribution, calcall it $\pi$.
For a chain with a stationary distribution (aka a marginal distribution that doesn't change depending on what time point you're at), $$ p(q_{t-1} \mid q_t) = \frac{p(q_t \mid q_{t-1})\pi(q_{t-1})}{\pi(q_t)} \tag{1}. $$
This is the definition of a reversible Markov chain. If you multiply both sides of the above by the denominator on the right hand side, you will get the more familiar definition of reversibility: $$ p(q_{t-1} \mid q_t)\pi(q_t) = p(q_t \mid q_{t-1})\pi(q_{t-1}) $$ which says being in state $q_t$ and then $q_{t-1}$ a moment later has the same chances as being in state $q_{t-1}$ and then state $q_t$ a moment later.
You'll notice that sub-scripting elements of the state space with a time index isn't great for expressing this idea very well.