If your work is closely related to the work of Professor A in the department, that should obvious from your research statement. The committee will likely give much more weight to research you have already done and describe in your statement then they will give to plans for future research.
So the first step is to make sure that you write the research statement in a way that directly shows how what you have done is related to Professor A.
Once that is done, you can also mention Professor A in your cover letter, and indicate that your research statement shows how your work is related. The committee can interpret that mention however they like. They may ignore it, they may view it as a sign that you would fit into the department, they may ask Professor A to look at your research statement and give an opinion.
If you can follow that advice, it can't hurt you to mention someone who you can work with. But only do that if there really is a clear relationship between your work and theirs. I have seen some job applications where the applicant claims they could work with someone, but there is no evidence on their vita or research statement that they can.
For example, they may have studied one subfield of a general field, and mention someone who works in a completely different subfield. In that case, it is not clear that the two could work together easily, and the claim in the cover letter comes off as exaggerated.