I'm not sure, but you can probably sign up for a QF frequent flyer account, and then use their website to search for availability. If you see some, then call the CO partner reward desk, and hope that they'll give it to you (I think they should, but don't know if they make all award availability available to all partners).
The alternative is to sign up for an
Expertflyer subscription, and you can search there for "U" inventory, which is for business class awards.
Keep in mind that there are a number of QF flights to check: SFO-SYD, LAX-SYD/MEL/AKL/BNE (and also JFK-SYD, but you're probably better off routing via the west coast). I think I've heard that AKL and BNE are generally easier to get availability. I'm not sure if the CO award allows onward travel within Australia, but I would just focus on getting
TO Australia/NZ, and then you can always buy tickets for the intra-oz flights.
Also, try searching in the QF forum for more tips and hints, and there are probably some posts in the AA forum as well. Good luck!