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As long as it's all on one ticket, then as explained on this Qatar airways page, the most significant carrier's allowance applies
Since the bulk of your journey is on Qatar airways, including at least one long international flight, the Qatar ticket's allowance of 2 bags will apply throughout
Additionally, since both airlines are part of One World, you'll be able to check your luggage through the whole way. Leaving America, you'll be able to check your bags at the starting airport, and not see them until after your final flight. Coming into America, you'll be able to get them tagged to the final location, but you'll need to collect them at your first American airport + clear customs with them + hand them to AA bag re-drop after customs to carry on to the final hop