"I-" Come on, get your head back in gear, M'gann. "I...I'm sorry, I should introduce myself?"

That might make this slightly less awkward, at least.

"I'm M'gann M'orzz; Miss Martian and J'onn J'onnz's apprentice. You know about him...Right? The Martian Manhunter? In the Justice League with Superman and the others?" Maybe that would help set some common ground down, at least, as she watched this Kara-person carefully. (And reminded herself to move forward with the line.)