No, before playong the game, all players must go out to destroy a civilization. Afterward you can give start player token to the one who accomplished this task last. The slowest one?
Technically you only need to have one player who has destroyed a civilization, and it need not be immediately before the game. If only one player has destroyed a civilization then it doesn't matter when that occurred -- even if the player was a baby when they destroyed the civilization! (Serious gas, maybe?)
But why take chances? If you're a serious Gentes player, you should make a point of destroying civilizations at every opportunity. And in particular, you should schedule time to do so just before playing in a Gentes tournament!