No, you can EV train any Pokemon that doesn't have maximum EVs yet. That includes newly hatched Pokemon or any Pokemon you just caught in the wild.
If you have been using a Pokemon for a while, they will have gotten many EVs from battles already, but you can remove those EVs with special EV-reducing berries: Pomeg Berry (HP), Kelpsy Berry (Attack), Qualot Berry (Defense), Hondew Berry (Sp. Attack), Grepa Berry (Sp. Defense), Tamato Berry (Speed).
In X&Y you can also use a Reset Bag in Super Training which resets all EVs.
See our EV section for more details.