As I recall, the Picard character did lay some French stuff on us very early in TNG, referring to a fellow ship captain as "mon ami" and also waxing nostalgic about the order of the colors blue, white, and red in their appearance on the French flag.
I don't remember him ever saying "mon ami", but I'll take your word for it. At least we had "Q" throwing "mon capitaine" at him all the time. Also, I think "Q" did toss a few other French words his way, hoping to coax a little French of him. But Jean-Luc never takes the bait.
I did find this (don't know how accurate it is, because it had horrible spelling mistakes that I corrected):
Captain Picard claims to be French, but is actually English due to the Globalization of Earth. In the twenty-fourth century there are only two countries, USA (consisting of North America, the European mainland, Upper, Middle, and Lower East, Asia, South America, and Antartica)* and its dependent protectorate of Great Britain (consisting of Great Britain.)
That seems more plausible to me... although I'm sure anybody on TBBS who is French born-and-raised would be insulted at the idea that anyone French would allow their language and accent to fall by the wayside.