No, you don't need to be a member of the LDS church to access their records or use their facilities at the FHC. They will have mixrofilm and microfiche readers and sone cebtres even have printers. Best to take tiur camera if you'd like a copy. The FHCs are all staffed by volunteers and they are very helpful and friendly. You'll need to order your films beforehand and they will email you once your film has arrived (I'm in Australia too and they take around a month or so to arrive after ordering microfilms online).
There are several FHCs in Brisbane - go to their website and work out which one is closest to you (sorry, I'm a long way from Brisbane so can't answer which is the best!) : https://familysearch.org/locations/centerlocator?c=City&...
Good luck with your research!