The phrase kia ora is a Maori greeting and the trademark attempt has sparked anger from the indigenous community.
The airline said it wants to trademark the magazine logo, not the phrase itself.
The New Zealand Maori Council described the move as "cultural misappropriation" and "frankly despicable".
The council is a statutory body set up to represent the Maori community.
"My warning to Air New Zealand is I sure as hell will put a stop to this," head of the council Matthew Tukaki said.