Hi Connorballs,
Yes that is correct, you as parent will have to apply for a name change on your son's behalf. This is the link to the application you have to fill out to register a name change for a minor in NSW. It is for children born in NSW or born overseas and having lived in NSW for 3 years.
http://www.bdm.nsw.gov.au/Forms/74-CON-Child-NSW-30May2014.pdf
This application is to be signed and consented to by both parents. However if you are unable to obtain the consent of your ex, you may need to take other steps. You might need to undertake mediation with the ex, and if that fails obtain a court order. Courts always place interests of the child as paramount in determining any issue. It will consider all short and long term effects of changing the child’s surname, embarrassment likely to be experienced by the child if its name is different from that of the parent with residence or day to day care of the child; confusion of identity, effect on relationship with a parent, contact that the other parent has had and is likely to have in the future with child, degree of identification that the child now has with the other parent and any step parent and the desire of the other parent that the original name be retained.
I suggest contacting the registry of births deaths and marriages to discuss your options.