Applying to Teaching Position in the Same Department as My Advisor... Should I Mention Her?
I would mention her, certainly. You don't need to make a big deal of it, but just mention somewhere that you worked on your dissertation under the guidance of Prof M. It is probably a matter of public record in any case. 

You should also probably give her a note that you are applying to her department. Under the circumstances you state there should be no issue about it, and she may be able to help you along if your relationship was good. Being at the same place gives you a way to carry on research trajectories already set in motion. 

She probably won't do for a formal letter of recommendation, however. 

And you can, of course, ask her before you apply about the advisability of it and also this very question.