The OQLF favours two words for hamburger: un hamburger and un hambourgeois, and they recognise an informal usage: un burger.
The thing about words favoured by the OQLF, though, is that they aren’t necessarily the words in use. Hambourgeois was created to replace hamburger, but it never took on. So, in actual usage, we really only have:
- un hamburger
- un burger
Use caution if you consult the Grand dictionnaire terminologique. It doesn’t reflect how French is really spoken in Québec — it reflects how the OQLF would like to see French spoken in Québec. It’s a collection of a) words in use that they approve, b) words in use that they disapprove, and c) words that are used little or not at all but that ideally they’d like to see catch on, like hambourgeois.
A hamburger restaurant called Harvey’s has this as its slogan in Québec:
à chacun son burger
À chacun son burger literally means to each his (own) burger, but more naturally it means something like your burger, your way. That’s because you can choose what you want on your burger at Harvey’s.
Burger is pronounced as in English (beurgueur), with English r‘s and all. It still uses normal French stress, though (i.e., stress on the second syllable).