My cousin has a New Zealand green card, so I can apply for relative immigration.

Immigrants are family units, including spouses and children, and generally speaking, the age of children is also limited, and the restrictions in different countries are different. Children over the age of 2/kloc-0 in the United States cannot immigrate with their parents and must apply independently. Your cousin's relationship with you is obviously not a direct relative, so you can't immigrate to New Zealand through relatives.