According to a video tweeted by the Guam News on Friday, Calvo while talking to Trump via an office phone switched the call to speaker phone to enable viewers hear the conversation clearly, Xinhua news agency reported.
"I just wanted to pay my respects and we are with you one thousand per cent, you are safe. We are with you 1,000 per cent. I wanted to call you and say hello, how are you?" Trump said.
"Mr President, as the Governor of Guam, representative of Guam's people, and as an American citizen, I felt never more safe and confident with you at the helm," the Governor said with a smile.
"Don't worry about a thing," Trump said. "I have to say, Eddie, you're going to become extremely famous. All over the world they're talking about Guam and they're talking about you," the clip, about 45 seconds long, showed the President as saying.
Tensions between North Korea and the US escalated this week amid increasingly heated rhetoric between Washington and Pyongyang.
North Korean military on Thursday said it would prepare a plan by mid-August to strike Guam, a US island territory in the Western Pacific, with intermediate missiles.
It was a response to Trump's unusually stern warning to North Korea, when he said: "North Korea best not make any more threats to the US. They will be met with fire and fury like the world has never seen."
Trump on Friday furthered the rhetoric, saying the US was "locked and loaded" to deal with the nuclear issue on the Korean peninsula.