No.

.... or at least I very much doubt it. Originally, QF flyers didn't earn any miles on JQ, though they do now on certain fares. JQ is not a member of OneWorld and you cannot fly JQ on OneWorld fare products. Said another way, why would you want to fly JQ?

QF hived off a lot of its leisure/non-premium routes to JQ. It originally had very tight seat pitch, no assigned seating and it doesn't transfer bags (so even if you're connecting to a QF flight, you have to schlep your own bags).

Be aware also that certain QF fares (certain Red-E Deals) don't earn miles. The QF website makes it very hard to determine what fare class you're buying (you have to look at the source code of the webpage to figure it out). Most discount fares on QF only earn 50% miles on AA. All in all, caveat emptor.