If you were looking at coming up to the Gaspe for Atlantic Salmon fishing and wanted to use your spey rod the best time would be the first 3 weeks of June. That is when the fish begin to run and the water levels are high enough to warrant a spey rod.
Newfoundland rivers, in my experience, get salmon runs later in the season as you indicated. And, the rivers are much bigger than Gaspe rivers.
If you consider Quebec you do not need the services of an outfitter. They are helpful, though, especially on new waters. You can get a list of them on www.fpq.com
In Quebec outfitters are the only people who can offer guiding and lodging packages; make sure you hire a licensed outfitter.
I prefer the Gaspe because of the large salmon and clear waters. Newfoundland has lots of fish, though.
Best of luck!