The 74-year-old told BBC Radio 4's Front Row programme she would make a "terrific subject" but he prefers to paint people he knows.
"When I was asked I told them I was very busy painting England actually. Her country," he said.
An exhibition showcasing his landscape work is to be presented at the Royal Academy of Arts in London next year.
Speaking at the London launch of David Hockney: A Bigger Picture, the artist said: "I generally only paint people I know, I'm not a flatterer really.
"I've been requested and it's actually a terrific subject, but I require quite a bit of time."
The Hockney exhibition, which runs from 21 January to 9 April, will be one of the countdown events to the London 2012 Festival, the culmination of the Cultural Olympiad.
With works spanning 50 years, it will explore the artist's fascination with landscape.
Inspired by his native Yorkshire, many of its large-scale paintings will have been created specifically for the exhibition.
The works will be shown alongside related drawings and films.
The artist, who was born in Bradford, said he had returned to paint in Yorkshire because "it is a landscape I know from my childhood and it has meaning".
"I never thought of it as a subject until 10 years ago when I realised that at my age that it is a terrific subject, a marvellous place.
"I love looking at the world, there is an intense pleasure from my eyes. Enjoyment of the landscape is a thrill."
The exhibit will feature three groups of new work created since 2005, when the artist returned to live in Bridlington, East Yorkshire, which use a variety of media.
A series of films produced using 18 cameras will also be displayed, on multiple screens.
"We filmed on a quiet road and no one never ever stopped us," Hockney said. "It is unique there because there are not many people.
"You can drive along the road in a car and not see anyone. It is a lovely little bit of England that is not spoiled."
The artist has embraced new technology in his recent works, using iPhones and iPads as tools for making art.
A number of his iPad drawings will also be on show at what will be the first major UK exhibition of his landscape work.
Mark Lawson's interview with David Hockney can be heard on Front Row later at 19:15 BST.
Artist David Hockney has revealed he turned down a request to paint the Queen because he was "very busy".