Do they get handed over to the new president or do they get shuffled back into the quorum card deck?
Good question. I don't think the rules or FAQ explicitly address this. The cards certainly don't get shuffled back into the deck; I don't think there's anything to suggest that would happen. The question is whether they are discarded or if the hand passes to the new President. The base game rule book really only has one statement that addresses this question:
Whichever player has the President title has control over this hand of Quorum Cards.
This implies the hand of Quorum cards doesn't really belong to the President, rather the current President simply has control over the hand. So if the title transfers to another character, nothing happens to the hand, but control of the hand belongs to the new President.
The Pegasus rule book offers some confirmation of this. Regarding execution, it says:
The player whose character was executed discards his hand of Skill Cards. (Quorum Cards in his hand are unaffected.) He also discards any Quorum Cards played on his character (for example the “Assign Mission Specialist” Quorum card).
And this about revealing:
The player discards down to three Skill Cards (or to zero if he has been revealed as a result of execution). He also discards any Quorum Cards played on his character. (If the player was President, the hand of Quorum Cards is not discarded. It is given to the new President instead.)
Since the Quorum hand is not discard as a result of the President being executed or revealed, then it is not discarded as a result of the President title being transferred to another character (since transfer also happens with execution or reveal of the President).