The suit filed Friday names 11 tobacco companies and covers a 60-year span from 1970 out to 2030.
The money is being sought under legislation passed three years ago in Quebec authorizing the province to seek damages to cover the costs of treating smoking-related illnesses.
The Canadian Broadcasting Corporation said Saturday Quebec was the fifth province to file such a suit.
The tobacco industry is not taking the challenge lying down. Imperial Tobacco Canada has its own lawsuit against Quebec. The company called Friday's action a "cash grab" by the provincial government it said was "a senior partner in the tobacco industry for decades."
"Governments have licensed us, have taxed us and our consumers, and have regulated us, all in full knowledge of the risks associated with tobacco use," Donald McCarthy, vice president of law for Imperial Tobacco Canada said.