Unprompted, Trump then offered this assessment: "Eddie, I have to tell you, you've become extremely famous".
White House Chief of Staff John Kelly also was on the call with Trump, Cooper reported.
Tensions have escalated between the USA and North Korea as the communist regime continues testing and expanding its nuclear program, reporting this week that it can now fit nuclear warheads on missiles.
"Mr. President, as the governor of Guam...and as an American citizen, I have never felt more safe or confident with you at the helm", Gov. Eddie Baza Calvo (R) said, according to a video posted to his Facebook page. "All over the world, they're talking about Guam, and they're talking about you, and I think tourism, I can say this, your tourism, you're going to go up like tenfold with the expenditure of no money, so I congratulate you".
"You know, they're showing so much, it's such a big story in the news and it just looks like a attractive place", he says.
"It's paradise", Calvo said.
And he told the former hotel magnate, "We have 95 percent occupancy and after all of this stuff calms down we're going to have 110 percent occupancy".More news: Premier League 2017 odds: Chelsea among betting favorites on opening EPL weekend
Referring to an "imminent missile threat", it includes advice like: "Do not look at the flash or fireball - it can blind you". "In time you will be able to leave a fallout shelter".
Trump says he has yet to speak with the governor of the territory, but says, "I feel that they will be very safe, believe me".
"So, with all of the criticism over there ..., we need a president like you", Calvo said.
Trump opened the conversation by telling Calvo, " Good morning, good morning.
Earlier, President Trump said the USA military was "locked and loaded" to deal with North Korea, should the need arise, as the countries' leaders increased bellicose rhetoric this week.
UNDATED (AP) - Officials in Guam are disseminating fact sheets to help residents prepare for a possible missile strike from North Korea.
If correct, and if the United States allows the missiles to launch in the first place, that leaves 17 minutes for tracking stations on land and sea - including the Pine Gap facility near Alice Springs - to calculate, target and try to destroy the missiles.