Margaret Rhodes said the monarch, who holidays at Sandringham in Norfolk, can imitate the distinctive local brogue "beautifully".
The Queen, who turns 90 this week, is still "full of laughter", Mrs Rhodes told BBC Radio 4.
She also has a sense of humour when things go pear-shaped, and enjoys telling amusing tales, her cousin said.
"She can tell very funny stories of things that have happened to her where things might have gone just a fraction wrong," Mrs Rhodes told the PM programme.
One such tale might include the time she met a shopper in Sandringham who did not recognise her and declared that she looked just like the Queen, prompting the monarch to remark: "How very reassuring".
Ms Rhodes said the Queen's vocal talents were not confined to the East of England, and years spent at Balmoral in Aberdeenshire meant she is also able to do an impressive Scottish accent.
Follow updates on this story and other news from Norfolk
A guide to the Norfolk accent
Source: Friends of Norfolk Dialect
She will spend her birthday, on 21 April, at Windsor Castle, where she will unveil a plaque on the Queen's Walkway.
Gun salutes will be held at all saluting stations across the UK at 12:00 BST, and members of the Army cadet force will take beacons to the top of the highest peaks of England, Scotland, Wales and Northern Ireland.

Riot police quelled skirmishes between rival fans behind one of the goals just before kick-off in England's 3-0 win.
An English FA statement said the incidents had not taken place in the area where fans who bought tickets from the FA were located.
The cases will be dealt with on Thursday, 22 October.
"We would encourage the authorities to identify anybody involved in disorder and deal with them appropriately through the criminal system and with banning orders," the English FA said.
"We will, of course, fully cooperate with Uefa."
When fighting broke out, local police quickly intervened to form a barrier between fans to prevent the violence escalating.
Some England fans appeared to have purchased tickets for sections of the ground designated for home supporters.
Lithuania's FA has also been charged by European football's governing body over breaches of ticketing rules and for having blocked stairways.
BBC Radio 5 live reporter Alan Green reported he had seen missiles being thrown by fans.
And BBC Sport's chief football writer Phil McNulty added there had been "a few brief skirmishes between rival fans at one end of this tiny stadium where some England fans have posted themselves near Lithuania supporters".
England won 3-0 with goals from Ross Barkley and Alex Oxlade-Chamberlain, either side of an own goal from Lithuania keeper Giedrius Arlauskis.
The victory meant Roy Hodgson's side won all 10 of their qualifying matches for next summer's European Championship in France.
The Queen is a "very good mimic" with a knack for imitating regional accents, her cousin has said.
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Uefa has charged the English and Lithuanian Football Associations over crowd disturbances at their Euro 2016 qualifier in Vilnius on Monday.