Probably. But even if BA can't, you should still be able to use the airside Flight Connections route to go straight to T3, and pick up a BP at the transfer desks there.

However, if you can OLCI on the QF website and print a BP, then you don't even need to stop to do that.

If things change and you end up having checked baggage, BA will through-check it to the QF flight despite it being on a separate ticket, because QF is a OW airline and therefore within BA's policy for through-checking on two separate tickets.