Gaspésie in June
The bulk of the run in most Gaspésie rivers is later than June. A notable exception is the Causapscal which has an early run of big fish. It will cost you a fortune to fish it though, providing there is some space left. The Matapédia will cost less but it is a big river and rarely fished from other than a boat. There is a good chance of getting some big sea-run brook trout in June though. Try the previously suggested rivers for that (Dartmouth, York, Cascapédia).