The Berellians were a species known to Starfleet by 2368.

The Berellians had a reputation for being very inept in dealing with machinery, as Christopher Hobson claimed that no one would think of making a Berellian an engineer just as no one would think of making a Klingon a counselor. With these examples he wanted to prove his point – that an android like Data would not make a good captain – when Data was given temporary command of the USS Sutherland. (TNG: "Redemption II")

This species was only mentioned in dialogue.
In a deleted or unfilmed scene from DS9: "The Abandoned", Mardah mentions a dom-jot game that Jake Sisko played against a Berellian. (Star Trek: Deep Space Nine Companion - A Series Guide and Script Library)
The Star Trek Encyclopedia (4th ed., vol. 1, p. 78) identified the Berellians as a Federation member.