I've noticed this in a few episodes now, and have not been able to figure it out. The screenshot below, with a circle added to highlight the flag, is from Star Trek: The Next Generation - Season 5, Episode 26: Time's Arrow, Part I.
What is this flag?
It appears to be just a plain, white flag. It's possible that it contains some fine-lined designs in a light or reflective color, but I haven't seen any. I would more expect there to be a white-on-blue or blue-on-white flag that has the United Federation of Planets emblem, but this is clearly not it. The flag makes appearances in other episodes where events take place at or near the Academy, but it is still similarly blank-looking in all that I've seen so far.
So, what is it then?