The image, reported by the Sunday Mirror, shows a screen with the Prime Minister sitting in front of a laptop in the No 10 library.
One of his colleagues is draped in tinsel.
Downing Street said the Conservative Party leader had “briefly” taken part “virtually” in the quiz.
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The Sunday Mirror reported the event happened on December 15.
The festive event is said to have taken place only three days before an alleged rule-breaking Christmas party in No 10, which is currently being investigated by Cabinet Secretary Simon Case.
Deputy party leader Angela Rayner said: “While the rules said that people shouldn’t have Christmas parties at work and Britons across the country were doing the right thing, Boris Johnson was instead happy to preside over a culture of disregard for the rules at the heart of Government.
“Despite repeated denials of parties in Number 10, it now transpires that there were numerous parties, gatherings and the Prime Minister even took part in a festive quiz.
“Boris Johnson really believes it’s one rule for him, another for everyone else.
“He is a man unfit to lead this country.”
Official guidance at the time stated: “Although there are exemptions for work purposes, you must not have a work Christmas lunch or party, where that is a primarily social activity and is not otherwise permitted by the rules in your tier.”
At the time of the quiz, London was in Tier 2 which said there should be no mixing of households indoors, apart from support bubbles, and maximum of six people outside.
The Sunday Mirror quoted a source who claimed many staff were huddled by computers, conferring on questions and drinking alcohol while the quiz was taking place.
It reported Mr Johnson surprised staff by turning up on screen as quiz master for one round lasting between 10 and 15 minutes.
A No 10 spokeswoman said: “This was a virtual quiz.
“Downing Street staff were often required to be in the office to work on the pandemic response so those who were in the office for work may have attended virtually from their desks.
“The Prime Minister briefly took part virtually in a quiz to thank staff for their hard work throughout the year.”
The quiz reports come as Mr Case, who is also head of the Civil Service, investigates three alleged rule-breaking Government gatherings last winter.
Staff reportedly held a Christmas bash in No 10 on December 18, with a leaked video – filmed four days after the alleged drinks – showing senior Downing Street aides joking about a “fictional” party.
A second reported Downing Street event – an aide’s leaving do which is said to have taken place on November 27 – was allegedly attended by the Prime Minister, who the Mirror said made a speech.
Mr Case is investigating both alleged events, along with a festive celebration arranged at the Department for Education, which officials have expressed regret over.
The Department for Work and Pensions (DWP) has since confirmed that staff working for Therese Coffey drank alcohol and ate takeaways “late into the evening” on a number of occasions while coronavirus restrictions were in place.
It comes after the Sunday Mirror reported that political staff and officials frequently drank after work until the early hours of the morning, and ordered food to the Work and Pension Secretary’s office at the department’s Whitehall headquarters.
The DWP has confirmed there were times when alcohol was consumed in a work space outside the Cabinet minister’s Whitehall office but stressed it took place while work was continuing past normal employment hours.
A source told the PA news agency the desks in the office are socially distanced and there was “no party atmosphere going on”.