"I have to say, Eddie, you're going to be extremely famous", Trump said.
The governor praised Mr Trump's handling of the latest escalation of tensions with North Korea, that saw Pyongyang threaten last week to target Guam, a United States territory, with four ballistic missiles.
Calvo posted a video of the call to his Facebook page, as well as YouTube.
"Don't worry about a thing", President Trump responded.
US President Donald Trump waits to speak on the phone with Irish Prime Minister Leo Varadkar to congratulate him on his recent election victory in the Oval Office at the White House in Washington, DC, on June 27, 2017.
"I have to say, Eddie, you're going to become extremely famous". As part of the short conversation, Trump also assured Calvo that he had become "extremely famous" as a result of Guam being in the press and said that tourism in the US territory would go up "tenfold". All over the world they're talking about Guam and they're talking about you. "It just looks like a handsome place", Trump said. "It just looks like a handsome place", he said.
Calvo concurred, giving Trump a tourism pitch and describing it as "paradise".
"Mr President, as the governor of Guam, representative of Guam's people, and as an American citizen, I felt never more safe and so confident with you at the helm", the governor said with a smile.
North Korea has announced a detailed plan to launch a salvo of ballistic missiles into the seas around the US Pacific territory of Guam, a major military hub and home to US bombers.
It said White House Chief of Staff John Kelly spoke with Calvo separately during an earlier call. He doesn't talk so much more anymore and we'll see how it all works out.