"We've made it clear that we hope to resolve this through talks," Tillerson said. "I think the most immediate action that we need is to calm things down."
However, a day later Trump tweeted his view that Tillerson should "save his energy".
"I told Rex Tillerson, our wonderful Secretary of State, that he is wasting his time trying to negotiate with Little Rocket Man," Trump tweeted on Sunday. "...Save your energy Rex, we'll do what has to be done!"
He followed up later in the day.
"Being nice to Rocket Man hasn't worked in 25 years, why would it work now? Clinton failed, Bush failed, and Obama failed. I won't fail."
Little Rocket Man is Trump's nickname for North Korean leader Kim Jong Un.
In his interview with Fairfax, Brown said many New Zealanders didn't understand the true threat from North Korea.
"They understand, 'Gosh, we're so far away and it really doesn't affect us'. Well, it does if he decides to drop a hydrogen bomb in the Pacific Ocean.
"The fallout could come here. It could affect the fishing and all the sea life. It'd dramatically affect climate and economy and the ability to travel freely in that region without being contaminated. So, yeah, it does affect New Zealand."
Brown, a former Massachusetts Republican senator, was one the first ambassadors in the Donald Trump Administration to be confirmed, and arrived in New Zealand in June. He is a former male model and posed nude when he won Cosmopolitan magazine's "America's sexiest man" competition.