Transfer League of Legends account

Unfortunately the answer is NO, the servers have a different database for user accounts, additionally the two servers use two different clients.

If you want to play on the European server, you will have to download the LOL Client and choose Europe and create a new account.

EDIT: If you were asking for the existance of an Account Transfer between servers, this feature has been discussed a lot on the official forums but there has not bean a RED response on any of them.