"Vacancy" is a concept meaning availability of a location (hotel room, apartment, conference room, etc). So a hotel might advertise "vacancy" meaning that it has rooms available, and "no vacancy" if it has no available rooms.
If a location is "vacant" it means you can move into it or use it, because there is nothing in it right now.
P.J. is using the analogy of the girl "moving into" his heart, meaning she is living in his heart (that is, he cares for her romantically). Gabe is extending the "moving in" analogy and joking that P.J.'s brain has a "vacancy"—meaning someone would be able to move in there as well—because it is empty, meaning there are no thoughts in it. (See the expression empty-headed.) It is commonly said that being in love prevents you from thinking rationally.