Hi Shish87,
I am new to the forum as well! I'm currently in the process of applying for a PhD at the Victoria University in Wellington. I travelled to New Zealand last month to visit my partner's family and relatives (but I'm not Kiwi) and took the opportunity to go down to Wellington for a few days and chat with a Prof. about a prospective PhD there. I contacted her beforehand (via E-Mail) and explained my interest, topic, etc.
The chat went well and I'm now getting my documents together for the application in November. I am not sure about other Unis in NZ but Vic Uni has three PhD application deadlines March, July and November. They have a good website for prospective PhD students: http://www.victoria.ac.nz/fgr/prospective-phds/applying#deadline.
Do you have a potential supervisior in mind? Where would you like to go? Do you have a topic/subject you want to write about in mind?
If you are down there at the moment, I would recommend to get in contact with potential supervisors and go and have a chat with them.
All the best and enjoy NZ!