There are red deer quite close to Brisbane. AFAIK there's no closed season on them in Qld, nor any bag limit, but in Qld you can only hunt on private land, for which you need the landowner's permission.
Other options, a bit further afield, include chital, mostly centred around Charters Towers (a couple of days drive or a short plane ride north of Brisbane) but again only on private land and with permission.
You could also drive a couple of hours south into NSW and hunt fallow (for which the open season in NSW runs from 1 March to 31 October), and while in NSW you can hunt on public land you do need written permission, which is done through the NSW Department of Primary Industry (
http://www.dpi.nsw.gov.au/hunting).
You'd also need to sort out licensing for a firearm, if you're going to use one. It is likely you'll need a guide to organise permissions, licences and firearms.
Have you tried contacting the Australian Deer Association?