There was a point in the past where this might have worked. And with select pairings of airlines it might still work - depending on how well integrated the various airlines systems are. However I would say that it's unlikely to work on a Qantas/Qatar pairing.
This will almost certainly be a Qatar Airways ticket, with a Qantas leg on it. When you board your flight from Darwin, Qantas will notify Qatar that you've done so (they will "pull the coupon" for that flight - a term that comes from the days when they actually remove the coupon for that flight from your physical paper ticket - although it's all done electronically now days). They do this not just to let Qatar know you boarded the flight, but because doing that is the trigger than gets them paid when the ticket was booked on another carrier!
By the time you arrive in Melbourne, Qatar will know you didn't fly the first leg of your journey. It's possible they will not have actually canceled your connecting flight, but there's a real chance they will have. Even if they haven't, it will show up at boarding that you were not on the inbound flight.
Given you've said you might need to check a bag in MEL, you're definitely going to come to their attention - to the extent that I would say if you check a bag, your odds of this working are pretty much zero percent. If you don't check a bag you might be able to slide through, and/or try and lie and convince the Qatar staff you did take the initial flight - but your odds of it working are low.
There's can also be further issues, such as what happens if your DRW-MEL flight is delayed. In this case the airlines will potentially proactively re-route you onto a later MEL-DOH flight, or even onto a completely different route (eg, DWR-BNE-DOH or DWR-SIN-DOH).
One way or another, this is asking for trouble. I can't say with certainty you can't pull it off, but your odds are low, and you risk losing the entire ticket (including your return if there is one) if you try.
