If you’re a platinum frequent flyer with qantas you can phone them and they will submit a request for the dates that you want, and usually within about 60 seconds the system will come back with availability. You may need to give a range of dates you are willing to travel. This has worked for me on my last 3 international trips (Japan x2, and USA).
You may be able to do this as a gold or silver frequent flyer too (not totally sure though, as it’s a different call centre) but it’s likely there are less seats available. Even online there is more availability that shows for reward seats for Platinum frequent flyers than even gold.
I’ve always found it hard to find seats for 2 people online - you either have to book 11 months out and even then can be difficult, or just keep checking every day until something pops up - sometimes the routing may be strange too. You can also book it as two separate one way bookings - when you see seats available one way, book them, and then keep searching until the return comes up. If you’re away for a month or more, often the availability will come up one way but not the other, but by the time the return comes up, the original flight there has gone. This may cost a little more in taxes but can work to get the seats you need.
Other option are any seat/points plus pay bookings, but these are way more points (sometimes 5-10 times the points of a classic award) and are very poor value.