The ship caused a scene in the Marlborough Sounds when it sailed too close to rocks and was reported to Maritime New Zealand.
On Thursday, it will visit Wellington, where passengers will be entertained at Te Papa by the Modern Maori Quartet in an event hosted by actor Temuera Morrison. It will visit Tauranga on Friday.
The boat is considered small for a cruise ship and is billed as offering an intimate, luxury trip. It's 180m long and 25m wide, has nine decks and weighs 30,277 tonnes.
It has a pool, spa, five dining rooms, a casino and a gym.
A voyage on the Azamara Quest costs about $400 a day.
The ship caters to a mostly over-40s demographic, and nearly 70 per cent of the passengers are from North America.
In April, it will undergo a US$25 million makeover to give it a more contemporary look.