We consider classical-quantum (cq-)channels with memory, and establish that Arıkan-constructed polar codes achieve the classical capacity for two key noise models, namely for (i) qubit erasures and (ii) unital qubit noise with channel state information at the receiver. The memory in the channel is assumed to be governed by a discrete-time, countable-state, aperiodic, irreducible, and positive recurrent Markov process. We establish this result by leveraging existing classical polar coding guarantees established for finite-state, aperiodic, and irreducible Markov processes [FAIM], alongside the recent finding that no entanglement is required to achieve the capacity of Markovian unital and erasure quantum channels when transmitting classical information. More broadly, our work illustrates that for cq-channels with memory, where an optimal coding strategy is essentially classical, polar codes can be shown to approach the capacity.