From Burlington to the town of Gaspe is 600+ miles.
I'm not sure about public transportation in QC as I walked but the good stuff is right there where you get off the ferry.
A great source for local information is www.bonjourquebec.com
and you can order booklets about the various regions, my stuff arrived in days, pretty cool. You'll want the ones for Gaspesie, QC, and Charlevoix which is where Baie-Saint-Paul is located.
Other than Montreal I don't know Quebec that well so take this for what it's worth. When I did a tour a couple years ago the 3 areas that I found most interesting where, Quebec City, Baie-St-Paul and Cap-des-rosiers ( which is near Gaspe ) and in between it seemed like many hours of just riding along.
QC is about a 5 hour drive from Burlington but you will feel like you are in Europe; beautiful, romantic, and lots of old world charm. Not to be missed.
Baie-St-paul, in addition to being quite nice there is tons to do in the area; great hiking, whale watching, the Saguenay Fjord ( 2nd largest in the world ), Isles aux Coudres and some funky little villages here and there. Tourism is important to the region so there is a ton of things to do like Kayak, sail, horse back riding and so forth. It's a very good place to base camp at a B&B, do day trips, and finish the day with a great meal in town ( my favorite is L'orange ).
Cap-des-rosiers is a long haul to get there but quite nice once you are out there. Parc Forllion is stunning and Pierce Rock is breath taking in person. Of course there are alot of boating related activities too, hiking, etc. It's much more roughing it and rustic in nature. I've got some great contacts out there, only wish it where closer.
Me? Well next time I will either do a QC/Baie-St-Paul trip or a Cap-des-Rosiers trip, but doing it all in one ride was a bit much for me.
Other stuff. High season is mid-July through Laborday Weekend and you will want to have reservations for sure. Most of the tourists are Quebecors and it will help to know a little French. Some basics expressions and a good attitude works well. And everyone will love you if you pay cash.
Good luck and let me know when your coming out this way, it would be fun to meet up.
BTW, congrats on the 990, great bike.