The Queen spoke yesterday with Ms Simon, Canada's first indigenous Governor-General, via video link. A brief extract of their conversation was shared on social media by the Royal Family's official Twitter account.
The monarch appeared in very high spirits as she spoke from her home in Windsor.
To pay tribute to Ms Simon and her country, she donned a brooch shaped like a maple leaf, the symbol of Canada.
In the brief clip shared on Twitter, the Queen told Ms Simon: "It's very nice to have a chance to talk."
The Governor-General replied: "Yes Your Majesty, it's wonderful to be able to speak to you today."
Speaking about her appointment, the Queen told Ms Simon: "So you are taking over a very important job."
The Governor-General replied: "Yes, I am very privileged to be able to do this work over the next few years.
"I think it's vitally important for our country."
The Queen then, curious to get to know her representative across the Atlantic, said: "Indeed, yes.
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"As your work, your own work, you have been very busy."
Ms Simon replied: "Yes, I've devoted a lot of my time to working on Arctic issues, because I am from the Arctic, and also working to get a lot of things organised with the Indigenous peoples and the Inuit of Canada."
Ms Simon is the 13th Governor-General of Canada to be appointed during Her Majesty’s 69-year-long reign.
Her appointment was announced earlier this month after Canada's Prime Minister Justin Trudeau chose her and the Queen accepted his recommendation.
Upon being officially appointed, Ms Simon said in a speech held at the Canadian Museum of History in Gatineau: "I can confidently say that my appointment is a historic and inspirational moment for Canada and an important step forward on the long path towards reconciliation.
"Indeed, my appointment comes at an especially reflective and dynamic time in our shared history."
Ms Simon, an advocate for Inuit rights and culture, is an Inuk from Kuujjuaq, a village in northeastern Quebec.
In her role, she will act as Commander-in-Chief of the Canadian Armed Forces and a representative of the Crown in Canada's Government.
Ms Simon has extensive experience in diplomacy and government, having previously served on the board of the Northern Quebec Inuit Association, as Canada's first ambassador for Circumpolar Affairs from 1994 to 2004 and ambassador to Denmark.
Her appointment came five months after the former Governor-General, Julie Payette, resigned after an independent probe looked into claims of verbal abuse and bullying perpetrated by her.
The Queen has visited Canada on more than 20 occasions during her reign.
She last travelled to the country in 2010 with the late Duke of Edinburgh by her side.
On the following year, the Duke and Duchess of Cambridge paid their first visit as a couple to the country, touring Ontario, Quebec, Prince Edward Island, the Northwest Territories and Alberta.
They returned in 2016 with Prince George and Princess Charlotte, who took part in a garden party filled with balloons and treats.
The Queen is the head of state of the UK, Canada and 14 other realms.