If I am already a Qantas Frequent Flyer, can I earn Qantas Frequent Flyer points in addition to my Emirates Skywards Miles?

No. You can only earn either Emirates Skywards Miles or Qantas Frequent Flyer points per flight. The benefits you will receive will be as per the Frequent Flyer number quoted in the booking.

Related questions

Can I also redeem my Emirates Skywards Miles for Qantas flights?
If I fly on an Eligible Flight but use a frequent flyer number from another airline loyalty program, for example Qantas, will I still earn Starpoints on that flight?
Are customer benefits different on codeshare flights?