The cheapest tickets on Qantas' web site have NEVER earned me AA points over the last five years. They book into a non-earning class; it's likely yours therefore won't, but the fare rules page on Qantas when you select "red tail" or whatever it is should show you the booking class IIRC.
Eight or ten years ago I pulled down a whole bunch of miles (and Gold status) with a couple trips to OZ one year on cheap tickets, but I think those days are kind of gone.