OP, not sure if this is your first flight flying Aust domestic, but at least here in Aust, at the moment, we don't have a TSA or similar agency, or some agency demanding photo id for the next flight, or what ever the name of the US domestic flight pre flight sort of id checking is called.
Here, if you don't already have a BP you just have to go to a machine, or counter.
But if you have BP, the xray screener might ask you where you are going, but it seems to be idle chatter while they do a bomb residue test (random), but its not as bad as what you would be used to flying in the US.
Maybe in the future, more id checks would be required prior to Aust domestic flights, but right now, compared to other countries, flying purely Aust domestic flights is very easy.
Maybe the Aust government considers it too easy, they do have plans to let the AFP check id of people at airports, but at this stage its not law yet.
So if the US QF checkin staff have provided you with a QFd BP, that is enough for your Aust domestic flight.
If not you just have to go to a machine at T1 and print your BP and get to bag drop, drop off bag, and thats it.