You'll need to make sure you're a New Zealand tax resident. You do this by working out your tax residency status. This is different from your immigration status.
Tax residency status for individuals
In general, New Zealand tax residents pay tax to New Zealand on their worldwide income whether or not it is sent to New Zealand or been taxed in the other country or territory. New Zealand may give a tax credit for tax paid overseas.
Tax for New Zealand tax residents
Most people who become tax resident in New Zealand for the first time or return after 10 years away are eligible for an exemption from New Zealand income tax on most overseas income for around 4 years.
It’s important to understand if you’re eligible, when the period ends and what kinds of income are not covered by the exemption.
International tax can be complex and you may want to talk with a tax professional if you're not sure about your circumstances.