The following is an excerpt from the script of The Spirit of Christmas: Jesus vs Santa, made in 1995.
Cartman: Heh heh, yeah, Hanukkah sucks!
Kyle: Don't you oppress me, fatboy!
Cartman: Don't call me fat, buttfucker!
Kyle: Then don't belittle my people you fucking fatass!
Cartman: Goddamnit don't call me fat you buttfucking son of a bitch!
So no, Cartman didn't start hating Jews in the beginning of the show, he hated jews before the show even began.