We state and prove a quantitative version of the bounded difference inequality for geometrically ergodic Markov chains. Our proof uses the same martingale decomposition as  but, compared to this paper, the exact coupling argument is modified to fill a gap between the strongly aperiodic case and the general aperiodic case.
"A quantitative McDiarmid’s inequality for geometrically ergodic Markov chains." Electron. Commun. Probab. 25 1 - 11, 2020. https://doi.org/10.1214/20-ECP286