According to Yevamot 102a (Rava says that midoraita, a convert can judge a case between converts) and Choshen Mishpat 7:1, a ger can serve as a dayan in gerus cases.
Interestingly, a patrilineal Jew who becomes a halakhic Jew through conversion also can serve as a dayan in cases involving a born Jew, according to Choshen Mishpat 7:1. The Aruch haShulchan there says davka that someone with a Jewish father who then converts (zera yisrael) can sit on a bet din involving cases with born Jews, since that is enough to satisfy the requirement of “mi kerev achecha.”