JAL's availability on QR has been rather variable over time. In a period they usually had excellent availability, then basically same as other partners, and currently they do seem have very limited access.

I have redeemed on QR J to some nice destinations, but for my next trip it was literally impossible to find the matching availability with AA that my co-traveler found and booked, so I had to get a cash ticket.

No specific other tips than use AA to search, the try to see if JAL has the availability. The deterioration of the availability on JAL seems to more or less coincide with QR going to AVIOS. I wonder if there is a correlation or just a coincidence.