As we mentioned, geirim cannot be appointed to positions of srarah. We will go into more detail in the future regarding which appointments, and what about the son of a ger.
Shmaye & Avtalyon, were the Av Beis Din and Nasi. The Tumim (Choshen Mishpat 7:1) shows from Eduyos 5:6 that they were on the Sanhedrin, but I think a bigger proof is Chagiga 2:2 that one of them was nasi and one of them was av beis din. They were the teachers of Hillel and Shamai, and were either descended from geirim, or, as the Rambam (Eduyos 1:3) and in his introduction to Yad Hachazaka says, they themselves were geirim, and they had accents which led to them saying malei hin instead of hin - see Shabbos 15a and Eduyos (loc. cit.). (Tashbatz [see below] shows they were geirim from Yoma 71b and Sanhedrin 96 [it's interesting that the Tashbatz calls Perek Chelek "Aggada in Nezikin,"] Chagiga 2nd Perek [which I quoted], Shabbos 15 and Eduyos 5:3 & 5:6. Sanhedrin & Yoma do not seem to be clear proofs in my opinion.)
If we say they were converts how could they have been on the sanhedrin - that was srarah? The Tashbatz in Magen Avos, Avos perek 1 says (with a "shema" - maybe) that when there is no one as great as a ger, the ger may be appointed. The Tumim gives two answers: (he also asks that one may accept a ger as a judge in monetary manners but not in a capital case, because one may not forfeit his life, so accepting him does not help?) Even though a person may not forfeit his own life by accepting a ger as a judge, a melech may appoint a dayan to judge dinei nefashos, and Shmaye & Avtalyon were appointed by a melech of the Chashmonaim. His second answer is that they were on the sanhedrin, but they recused themselves from capital cases.