No, you'd have to get their permission first. And I would be REALLY surprised if they let you do that.
I think you've already answered your own question: TRADEMARKED LOGO. That means it's theirs to use, not yours
You cannot even use team logos in your own logo or players. A local casino got in trouble for using a NFL player in their promotions and got sued
I think it's common sense already that using any trademarked logo is illegal. In my opinion, some people are not really concerned whether it's legal or not to do so(they already know the answer to this) but rather whether they can get away with it or not.