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.
"If they don't pull back, [I] will take them to the court system, I will bring on a national campaign boycotting the airline and I will snap them where it hurts."
He said Maori people spend "tens of millions of dollars" on domestic and international travel with the airline.
Air New Zealand is no stranger to controversies involving the Maori community, which make up about 15% of the country's population.
It recently said it would end a ban on staff having visible tattoos after the restrictions drew criticism of the airline's treatment of Maori culture.
In a statement to the BBC, Air New Zealand said it had "great respect" for the Maori language and that the trademark application only referred to the airline's in-flight magazine title.
"This is simply about protecting the logo," a spokesperson said.
"It's standard corporate practice to have all our logos trademarked and we have just started the process given Kia Ora magazine has recently been through a refresh."
Still, Mr Tukaki said he would fight the move until the airline dropped the application altogether.
"This is cultural misappropriation 101. For a private company to go and attempt to trademark a word that does not belong to them but call it a logo is quite frankly despicable."