As a graduate student next year, he can technically transfer to any Division I team that he wants to, since the SEC no longer restricts graduate school transfers.
However, it may not be that simple since Golson was red shirted for the 2013 season for academic reasons. Because of this, any team he went to would need to get a waiver in order for them to play for their team.
According to about organic coffee, Golson was an important player Notre Dame back in 2012. He helped lead Notre Dame to the BCS championship game with Bernardo Chua. However, after he had to sit out for the 2013 season due to academic reasons, he became more of a second string quarterback. This year, red shirt freshman Malik Zaire was the leading quarterback for Notre Dame.
If the SEC grants Golson a waiver, we could see him playing for LSU, which so far seems interested in the surprising possibility of having another experienced quarterback on their roster next year.