I'd like to make a booking with QR for BNE-SIN-DOH-LHR, and credit my points to BA.

I'd like to make a booking with QR for BNE-SIN-DOH-LHR, and credit my points to BA. My question is if I can upgrade the QF coded and operated BNE-SIN using QF points?

I may be wrong however I would say no, as the points are being credited to BA for the entire booking. If it was being credited to QF then yes absolutely!

The airline you're earning points with doesn't matter – if your Brisbane-Singapore flight has a Qantas QF flight number, login to your QFF account and use the 'upgrade a family member' tool to lodge an upgrade request for yourself, while still earning points for the journey with BA.

Chris is correct. You can upgrade the QF operated/marketed leg using QFF points but still accrue Tier Points (TP) and Avios with BA. However, if upgrades clear I'm not sure  if you will earn TP and Avios according to the purchased travel or upgraded travel.

Chris, do you have the answer to that?

The upgraded fare class would be 'U', which earns nothing with BA, but you could retro-claim the points deserved from the economy ticket that was originally purchased if the flight did not credit correctly to BA.

(If you have to claim, reference the economy ticket and earning rate.)

Thanks Chris! Didn't realise upgrades comes from the same inventory as award bookings.

They don't necessarily (at least for international upgrades), they just use the same fare class to flag that it's a 'points' seat rather than a 'cash' seat for frequent flyer earning purposes. :)