The classical open relativistic Toda chain is a well-known integrable Hamiltonian system which appears in various different contexts in Lie theory and mathematical physics. As was observed by Gekhtman-Shapiro-Vainshtein, the phase space of the relativistic Toda chain admits the additional structure of a cluster variety. I will explain how this cluster structure can also be used to analyze the quantization of the relativistic Toda chain. In particular, we will see that the Baxter Q-operator for the quantum system can be realized as a sequence of quantum cluster mutations, which allows us to obtain a Givental-type integral representation of the Toda eigenfunctions, the q-Whittaker functions. Joint work with Alexander Shapiro.