A joint channel estimation and channel coding scheme is presented for finite-state Markov channels using polar codes. Unlike the conventional approach of first estimating all channel parameters and then performing channel decoding separately, the proposed scheme incorporates a subset of reliable estimates of channel parameters into the decoding algorithm and computes decoding metrics averaged over the statistical behavior of the Markov channel. By adapting list decoding (without any inner code), channel estimation and decoding steps can be performed iteratively to boost the reliability of channel estimation as well as error correction. In order to improve the performance even further, a new pilot transmission scheme is developed that utilizes the structure of polar codes and sends pilot symbols along with code symbols. This construction can be viewed as a new family of shortened polar codes that can be of independent interest. Simulation results demonstrate the benefit of the proposed approach compared to existing solutions.