To earn maximum miles on the BA Amex you should either book on BA.com or call BA. You can book the Qantas flights through both channels.

If you book through Qantas.com or call Qantas you will not get the maximum miles.

This is because Amex only sees the airline that sold you the ticket and the flight numbers, which appear on the statement. They do not see the operating carrier. So if it is sold by BA, you get 3x miles. If it is sold by any other carrier you only get 1.5x miles.

This works for other carrier bookings too. You can call BA and book a ticket on any airline in the world. The ticket will be issued on BA stock and sold by BA so Amex will see BA as the airline and you will get 3x miles. ^