In [Diaconis, Pang, Ram], the Hopf square map combines the key operations of a Hopf algebra, the multiplication and comultiplication, to give rise to standard Markov chains in probability theory, like shuffling and rock breaking. However, only commutative and cocommutative Hopf algebras are used. After summarizing their work, we expand upon it by using the Hopf square map on a non-commutative and non-cocommutative Hopf algebra– a q-deformation of a standard Hopf algebra that forms a true quantum group– to create a new Markov chain. Interpreted as a deformed walk on the natural numbers, we then analyze the chain and similar random variables, discovering a phase transition as the deformation parameter q is varied.