Layover in Canada with a DUI, will we still be able to use Canadian airport with no issues?

Pretty much what the title says. My girlfriend and I are planning a trip to Greece at the end of September till mid October. We both got our DUI's roughly 8 years ago. We both know we can't visit Canada unless we do a rehab program with proof, or do a temporary resident permit. We're leaving from Seattle WA and flying into Athens. With that being said, 90% of flights all have a connecting flight through Montreal Canada. We're worried that we're not going to be able to take those flights because it is Canada and we have DUI's. I have tried looking on Google for a couple hours and getting completely mixed responses. Finding a definite answer has been pretty much impossible. Since we're technically not visiting Canada will we run into any issues with customs while were at a canadian airport? So I came here to ask if anyone has experience with this? Or knows an answer to this question? Any info will help! Thanks.