Will I need a separate membership number to use on Qantas-operated flights?

No. When you book a Qantas-operated flight, enter your current Emirates Skywards membership number and any eligible Miles will be automatically added to your account, from March 31, 2013.

Related questions

What is a Gold Partner card?
When do I get e-gate access benefits?
If I am eligible for lounge access and departing from an airport with both an Emirates lounge and a Qantas lounge, can I choose either lounge?