As Rimouskois correctly indicated, the bulk of the run on the Gaspe' is (usually) in late June and early July.
Nonetheless, some rivers experience a superb early run: the Restigouche, Cascapedia, and the York are good examples. Although the fish are fewer and further between, they make up in size for what they may lack in numbers.
On the York in the 1st week of June, try "Petit Saumon"and "Baillargeon" in Zone 1 (Unlimited Sector); both very productive high-water pools.
In Zone 3 (reservation required), "Mississippi" and "Bluff" are outstanding during the early season.
In Zone 4 (reservation required), try "Still," another superb pool in high water. At that time of the year, the fish are usually found in the tail of the pool.
I will be on the York during your visit. I'll see you either at Tim Horton's for the pre-fish cup o' joe or on the river.