We investigate quantum compact groups which support quantum metric space structure. In our core example, we define an analog of the Hamming metric on the quantum permutation group $S_n^+$. The construction of our quantum metric relies on the work of Biane and Voiculescu. We also obtain an associated quantum $1$-Wasserstein distance on the tracial state space of $C(S_n^+)$. This is joint work with David Jekel and Anshu Nirbay.