The exchange took place in a brief and informal 30 minute audience at the Vatican today.
The queen and her husband, the Duke of Edinburgh, was on the ground in Italy for less than five hours before returning to the United Kingdom on a private jet.
It was the first time Queen Elizabeth met Pope Francis and she took with her on the plane home a gift for the royal grandson. Pope Francis handed the queen a precious stone globe with a cross on it for Prince George, who is about to depart on his first overseas trip to Australia and New Zealand.
The queen was given an antique parchment, which dates back to May 1679 with an "Urbi et Orbi" message by Cardinal Cesare Facchinetti. The pope traditionally gives a "urbi et orbi" message - meaning "to the city and the world" - at Easter and Christmas.
The queen gave the pope a hamper of what the Vatican said was "jams and drinks" produced on lands owned by the royal family. One of the drinks was a bottle of whiskey from Balmoral.
As a pair, the queen and the pope are leaders to billions of people around the world.
"If you think about it in terms of soft power then we're talking about 3 billion people because the pope is the spiritual head of more than 1 billion Catholics around the world, and the queen of course is head of the Commonwealth and that's 2 billion people," said the Italian author and journalist Barbara Serra.
This is the fifth time Queen Elizabeth II has met a pope in Rome, but this was her most informal visit ever. She wore no black veil, instead a lilac outfit and matching hat. By mutual request the was relaxed and "protocol-light," taking place in the pope's study inside the 1970's built Paul VI Hall, close to where he lives at the Vatican.
The protocols of this visit are markedly different to a state visit, which would normally see such a meeting take place inside the Apostolic Palace.
The only people present for this meeting were Pope Francis, Queen Elizabeth, the Duke of Edinburgh and a translator. Pope Francis speaks only "passing English" and is understood to have communicated via a translator.
The meeting with Pope Francis marks the 100th anniversary of the re-establishment of diplomatic relations between the United Kingdom and the Holy See. Historically though, the Church of England has difficult relations with the Vatican, as a break-away from the Roman Catholic Church.
Serra says Queen Elizabeth, who is 87 and Pope Francis, who is 77 are both charged with modernizing some of the world's oldest institutions. Serra says "that will probably be quite an interesting exchange between the two. I think the Queen has managed to do that (modernize) here in the UK. I'm not sure the Pope has had the chance to do it yet."
It is increasingly rare for the queen to travel abroad indeed this is her first foreign trip in three years.
The trip to Italy was planned for 2013, to meet the octogenarian President Giorgio Napolitano, who is said to have a friendly relationship with the British monarch. That trip was cancelled due to the queen's ill-health before being reinstated this year.
Today's tour began with a private lunch at President's official home, known as the Quirinal Palace. There was a group of 10 people seated for lunch, including the president and his wife, the queen and Duke of Edinburgh, along with the Italian Ambassador to the UK and his wife, the UK Ambassador to Italy and his wife, along with other senior level diplomats.
the UK and his wife, the UK Ambassador to Italy and his wife, along with other senior level diplomats.