I don't know what level you teach, but there are some schools for English students (at all levels, I believe). There are a couple English universities in Montreal - McGill and Concordia, and I think there are 5 or 6 English CEGEPs, which are post secondary schools that can either be used for technical programs or to prepare for university studies. If you teach school-aged children, you might want to check out the English Montreal School Board web site.
Do you think sponsorship is an option? If you and your partner have been together long enough or are married and are committed to each other long-term, it could be a way to at least secure your ability to live and work freely in Quebec (and then the licensing to work could be worked on separately).
I did find this on the Immigration Quebec web site:
Immigration-Québec - Teaching profession
I should probably add that Montreal is relatively bilingual, so an English speaker wouldn't necessarily have a problem, except perhaps regarding employment (but some people get by never learning). If you didn't want to live downtown, there are even English areas to live on the island of Montreal (the West Island, specifically).