You can. Best to search from the main cities that QR originates the long haul from. I.e. ORD-MLE on 22-FEB is showing seats in coach. If you find the flights you need, you can put them on hold, then call AA to try to add a points flight to get you to ORD from your home airport. This may be difficult, as the booking class for this positioning flight needs to be a certain fare code (I believe "T").
If you search from your home airport to MLE you will generally get no QR results (unless your home airport is DFW, ORD, etc)
You can use the calendar feature to more quickly search. Business seats seem hard to come by.
Also, there is a dedicated thread on FlyerTalk about this:
Qatar Airways / QR: Redeem AA Miles for Award (master thd)