Firstly, there is no First in Australia. We only have Business or WHY (although our Business Class is better than First on most US carriers domestically).
One option - sign up for a QF FF account - the cost to you with a US address will be $0.00 (plus 0¢ GST).
Once a QF FF member, you will be able to see J class award availability. Once you see that it is there, ring AA to book it (the availability is the same, regardless of whether you are an AA or QF member.
Having said that though, you will be pleasantly surprised by QF service in WHY. Unlike US carriers, you will be fed and watered in WHY - on some flights you will even get free beer (not that soft drink called Budweiser, but a real BEER) or wine in economy.
Seriously, I would be saving the miles and staying in WHY.
Dave