The US president has been in the UK since Monday for a three-day state visit and was welcomed by Her Majesty and the Royal Family.
As well as attending a private lunch with the 93-year-old monarch, Mr Trump was thrown a state banquet at Buckingham Palace on Monday night.
He told Piers Morgan on Good Morning Britain: “We had a great rapport. We had a conversation that lasted an hour and a half, non-stop.”
When pressed by Morgan about the contents of their discussion, Mr Trump said: “I heard that we are not supposed to do that [reveal what we discussed].
“I said tell me about this, I promise I will never tell anyone about it. I’m not going to break my promise in one day.”
'It was really something.'— Good Morning Britain (@GMB) June 5, 2019
We can imagine it'll be tough impressing @POTUS, but the Queen and Buckingham Palace seem to have done the job!
He says he and Her Majesty 'had a great rapport' and spoke 'non-stop' for an hour and a half.@realDonaldTrump | @piersmorgan | #GMBTrump pic.twitter.com/kjNk6WxVHG
He added he had “great respect” for the Queen.
It is considered a breach of protocol for heads of state to reveal what was discussed with Her Majesty privately.
Former prime minister David Cameron was left embarrassed in 2014 after he was overheard saying the monarch had "purred" when told the result of the Scottish independence referendum.
Shortly after the blunder, Mr Cameron said he regretted it and was “extremely sorry about it.”
He added: "It was a private conversation, but clearly a private conversation that I shouldn't have had and won't have again.
"My office has already been in touch with the Palace to make that clear and I will do so as well."
Donald Trump previously met the Queen during a business trip to the UK in July 2018.
The president and Her Majesty met for tea in Windsor and Mr Trump hinted they’d discussed Brexit during that meeting.
In another interview with Piers Morgan last year, he said the Queen had said “it’s a very complex problem,” but he did not reveal what she thought of Britain leaving the EU.
On his final day of the state visit, Mr and Mrs Trump will join Her Majesty and the Prince of Wales to attend a National Commemorative Event at Southsea Common, Portsmouth, to mark the 75th anniversary of D-Day.