Queenstown is surrounded by hills, so you'd have to get out of town a bit to see the Aurora properly. Or head down to Strahan on Macquarie harbour and Ocean beach which should provide better views. We are further north than Hobart and generally require a Kp of at least 6.5 to be able to see it.
The bigger problem will be the weather. The West Coast of Tasmania is renowned for it's rubbish weather. During summer we do have quite a few good days, but over on the east coast is more reliable.
For the record, I've been here for a bit over a year now and have yet to see the Aurora. The issue has been that every time the aurora forecast is good, the weather forecast is not