Australia doesnt really have livery yards like you are used to.
Most Australian Horse owners either have their own property or do agistment. Depending on where in Qld you are moving to you may find the local pony club does do it yourself agistment. Better to ask around once you get here as to what others do and look for something relatively close to where you live.
I live on 40 acres and have my own horses here and have one on agistment. Her owners provide her feed and all worming, I just feed it out with my own but they are friends so am not worried about them doing any of her day to day care.