When traveling on a codeshare flight, can I check in at either Emirates or Qantas counters?

No. If you are traveling on an Emirates-operated flight, you need to check in at the Emirates counter, and for Qantas-operated flights at the Qantas counter.

Related questions

Will Emirates Skywards members be offered additional checked baggage allowance on Qantas-operated flights?
What is the eBoarding Pass and how do I use it?
Do Emirates baggage allowance rules apply to my Qantas-operated flight?