There is no universally-accepted abbreviation for million or milliard. Neither the Trésor de la langue française nor the Petit Robert mentions one. The Québec Banque de dépannage linguistique does not mention any abbreviation in its article million et milliard, although it does give an example (but no rule) where “M$” is to be read “million(s) de dollars” in its general article about numbers.

You may occasionally find M for million and Md or Mrd or mrd for milliard, but those are rare, a lot rarer than M and even B in English. M more commonly stands for the SI prefix mega, which fortunately is the same number.

Billion is basically never used, even unabbreviated. France’s debt is written as “2200 milliards d’euros”, not “2,2 billions d’euros”.

Scientific publications obviously use SI prefixes in front of units. These prefixes are occasionally used to abbreviate amounts of money, but that’s far more common in English-influenced business documents than in the general press.