How to handle ego clashes within an academic department
The best advice anyone can give you, in my opinion, is: Don't get involved.

You're entering a new environment and it's in your best interests to just get along with everybody, as you already suggest you are trying.

If the issue pops up in any discussions, just avoid it. The last thing you want to do is get caught-up in a territorial dispute of which you probably don't want to know the origins, and that serves no purpose to anybody, least of all to you.

I would only advise discussing things with the Chair if these divisions start negatively affecting your work, and only discuss it in terms of how it affects your work. As you've pointed out yourself, chances are he/she can't do anything about it, or worse, he/she may be involved.

Things like this can happen even in the best departments, and the best thing that can happen in such a case is that new staff bridge whatever divide may be there by ignoring it completely.