I also have 2 middle names as is fairly common in France (are you French Canadian?). Same as yours, my green card is first name and initial of "second name" only.
But I've noticed that the NOA1 I just received has my full name with both middle names. (Funny enough, they added a line "care of" with just my first and last name in the address, ie Sally Jane Mary Smith c/o Sally Smith, lol.) I expect to see both my middle names on the naturalization certificate because of that.