The University of Otago has a large number of scholarships available for PhD study and a smaller number available for Master's study. Masters scholarships are only applicable for the thesis component of a Master's degree involving taught courses and a thesis, but there are some scholarships available for coursework Master's degrees.
Applications for PhD and Masters scholarships are made at the time of application for admission and there is no particular deadline.
More information about the University of Otago Postgraduate Scholarships can be found here.
There are also some New Zealand government scholarships for international PhD students, with an application deadline of 15 July each year. More information can be found here.
The Scholarships Office maintains a searchable database of scholarships available for study in New Zealand and around the world. Many of these are specific to particular areas of research or targeted at students with particular types of background.