A statement that defeats itself is called "self-defeating". For instance, the claim "All statements that contain the letter e are false" is self-defeating; it contains the letter e, so it is claiming that itself is false.
Is there a logical fallacy in this situation?
What you quote are two claims. Claims cannot be fallacious. Only arguments can be fallacious.
There are implied arguments here. The implied argument of "This group is judgmental" is "Being judgmental is bad, this group is judgmental, therefore this group is bad". This is valid logic, but arguably not sound.
The reply "Your saying the group is judgmental is judgmental" is responding to the implied argument. The implied argument is "You are saying that being judgmental is bad, but your statement is judgmental, so by your logic your own statement is bad". This relies on the further implied premise that all evaluations are judgmental. The term "judgmental" is generally understood to mean more than just than that. Furthermore, a statement being "bad" to say doesn't mean it's wrong.
If those defects are fixed, if a reply does explicitly assert that the first statement is making an argument, if the first statement is in fact making that argument, and if that argument contradicts the statement, then that would be a valid argument.