The kids in the program are more likely to be quirky themselves, and are more tolerant of quirks in others. I think they are especially tolerant of quirks in someone who is also able to contribute different and meaningful insights into class discussions, because they value that kind of input.
This has been our experience with a FT gifted program and an anxious/quirky child as well. I wouldn't say teachers actively accommodate, but for the most part they have been tolerant, and other kids are used to some quirkiness.