I think teams in the same league can make transfers soon after the last competive game is done but officially the transfer window opens in early June, although a lot of player's contracts end on the 30th June. Having said that, some clubs still do their business before that, they just don't make it official until 1 July. Case in point -
Edit: Having just checked. It opens on different days in different countries. So it'd probably be a bit of a hassle to open it early.