My grandmother was a Courté. Her grandfather was Edouard Courté born in Québec and married Prudente Duguay in Shippagan, NB around 1853. This is where the lineage dissappears.
However, there is a Michel Courtais listed in "The French Canadians 1600-1900" who married in 1795 in Paspébiac. So far, I am unable to link the Courtais name with Courté, although I am quite certain they are the same. Does anyone have any more information?