Not likely. But you will have to pay for discharge medications. I dont think you have to pay for any treatment as a public patient, but then you 'pay' in that you have to wait yonks to get treated.
If you go as a private patient in a public hospital, ie you chose your own doctor, ie the consultant, but you would be in a public hospital and can still have interns etc following around, then it is cheaper a bit.
If your are a private patient, the private health cover will cover theatre costs and stuff in network hospitals, ie most hospitals id say, but you pay for the excess, pathology and any other services that were used during your hospital stay, eg you pay for physio etc and the surgeon and anasethetists bills.