I did not miss the point that OP found that QF's search engine did not provide her/him with information which he/she wanted; however, I did provide clarifying information as to which carriers operate to the points which she/he has interest in.

It is not at all unusual for any particular airline's online booking tool to be incomplete, especially as to services not provided by that airline but by other airlines including by alliance partners. One simply needs to pick up the phone and call QF to make the reservation, but first one needs to determine on what flights one wishes to fly; to do so one needs to know what airlines have service to/from the desires airports.