The last time I was there transferring from a QF domestic flight to a JQ international flight with a 6-7 hour connection, the JQ counters weren't open until 3 hours before the flight. I met up with friends from Darwin during my layover so it was no big deal for me though. DRW is a very quiet place in between flights, so don't think QF would be open either. QF website may have more details under Airport Information for DRW.
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