No, neither you or your baby will be covered under medicare. You would certainly need to have good health insurance, and even then this may not cover your babies injections, you would need to find out exactly what they will cover you for. They usually only cover accidental or sudden onset illnesses.
"Medical services are excellent, but can be very expensive, so travellers should ensure that they have adequate insurance. Australia has a reciprocal health agreement with the United Kingdom providing for free hospital emergency medical treatment; proof of UK residence is required."