Yes, Australian first aid qualifications are recognised in New Zealand. Under the Trans-Tasman Mutual Recognition Act 1997 (TTMRA) a person who is registered to practice an occupation under an Australian law is entitled, after giving notice to the local registration authority for the equivalent occupation, to be registered and to carry on the occupation pending registration.
This means that a first aid qualification gained in Australia can be deemed broadly comparable to one gained in New Zealand. To maintain currency in New Zealand, a person holding an Australian first aid certificate need only attend a biannual refresher course. This must be done within two years and three months of either the original course or the last refresher course.