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

Do Emirates baggage allowance rules apply to my Qantas-operated flight?
Will Emirates Skywards members be offered additional checked baggage allowance on Qantas-operated flights?
Can I check in for Qantas flights on the Emirates website?