We study the C2 Ruijsenaars-Schneider(RS) models with interaction potentials of rational and trigonometric types. The Lax pairs for these models are constructed and the involutive Hamiltonians are also given. Taking nonrelativistic limit, we obtain the Lax pair of C2 Calogero-Moser model. PACS: 02.20.+b, 11.10.Lm, 03.80.+r