Although a motion was passed at the Monthly Council meeting on Monday night to write to her, it divided opinion in the Chamber.
Proposor, DUP councillor, Bert Johnston, said he felt the letter of congratulations would be fitting, given her intentions to visit the County in the coming weeks.
With speculation growing that she will open the new South West Acute Hospital on June 26, he told Sinn Fein Chairman, Thomas O'Reilly that he would have the opportunity to meet with her there.
It is understood Queen Elizabeth will also attend a church service in Enniskillen during her visit to the County town in a fortnight's time.
Pointing to her visit to the Republic of Ireland last year, Mr Johnston submitted that the Queen had reached out across the divide.
"I do know the Royal House is very much attached to Protestantism but I notice now that a hand is being reached out and indeed it is being grasped by people of a different religion and culture," he said, "I think it is good to see this happening and I do know that the Deputy Leader has indicated that he will be shaking her hand and has been involved in sending her a gift from the Executive at Stormont."
While his proposal was seconded by fellow DUP councillor, Alison Brimstone, Sinn Fein made their stance of opposition clear.
"If I weren't an Irish Republican I would still be anti-monarchy," Brian McCaffrey told Council members, "And so I want to be disassociated with this proposal."
Independent councillor, Bernice Swift, voiced a similar position.
But when the motion was put to a vote, the Unionist side of the Chamber was supported by SDLP councillor, John O'Kane, meaning Mr Johnston's proposal was carried with the majority of 10 to nine.
Councillor O'Kane's fellow SDLP members, Brendan Gallagher and Frank Britton chose to abstain.
UUP councillor Raymond Farrell said he was "delighted" that the motion had been tabled.
"I am also doubly delighted that gift from Belleek Pottery is being presented to her on behalf of the Stormont Executive, one because it is from Fermanagh, and two because it is from North Fermangh. I hope she has many years of pleasure looking at it," he said.