Emily Thornberry criticised Sky News presenter Dermot Murnaghan for "pub-quizzing" her as they discussed Brexit talks, saying it was "patronising".
The Labour MP declined to name French minister Jean-Marc Ayrault, adding: "Can we talk about some serious stuff?"
Mr Murnaghan said he had asked male politicians similar questions.
He went on to refer to the time in 2011 when he asked the then shadow chancellor Alan Johnson about national insurance rates - a question the Labour MP was unable to answer.
Ms Thornberry was asked if she had taken part in any Brexit talks with her would-be counterparts in Germany and France, and then if she knew the name of the French foreign minister.
"Don't start pub quizzing me, Dermot," she said.
She was then asked again if she was able to name the minister.
Ms Thornberry said: "No and I'm not going to start answering your questions on this."
She added: "Do you know what, what really upsets me about your attitude to me is that you do this with me. I don't remember you doing it with anybody else you know.
"Have you done it to David Davis? Have you asked these questions? Do Sky journalists have a go at Boris Johnson on this basis? How about Liam Fox? Do you do pub quizzes with them?
"I mean honestly. Can we talk about some serious stuff?"
Ms Thornberry suggested that Mr Murnaghan should ask her questions about the situation in Syria or North Korea.
Mr Murnaghan then asked her if she knew who the South Korean president was in relation to the recent news that North Korea had carried out a fifth nuclear blast.
"I'm not getting drawn by you into this nonsense," she said.
Mr Murnaghan went on to ask Ms Thornberry about anti-Semitism within the Labour Party.
She said Labour is committed to tackling the issue, as well as racism and sexism, whenever it occurs.
She said: "There is always more to be done and do you know what, there is certainly a lot more to be done by the Tories and I certainly think sometimes when it comes to sexism, some Sky presenters need to look at themselves too.
"It really upsets me that every time I come on here, you do another pub quiz with me because you do not do it with anybody else and I do think it's patronising."
Mr Murnaghan replied: "It's not, because you are the shadow foreign secretary and this was about the French foreign minister Jean-Marc Ayrault who Boris Johnson has met now several times."
Ms Thornberry said she wanted to "take this offline because I have got a lot to say to you and I don't think a lot of it ought to be broadcast".
She later told BBC Radio 4's Westminster Hour she had "a bit of history" with the Sky presenter.
"When I was first made shadow defence I remember getting loads of questions about 'what do you know about defence? You know nothing about defence'... it was as if I was needing to get permission to be heard.
"I don't think those are the sort of questions that get asked of Boris Johnson or David Davis or Liam Fox - we were three minutes into the interview and we were already getting 'never mind what Labour's policy is on anything, never mind about serious issues like Syria - let's have a bit of a laugh'."
She added: "I'm sure I should know the names of the foreign secretaries of all 192 countries but I don't."
The interview has sparked a wave of debate online.
Isabel Hardman, the assistant editor of The Spectator, tweeted: "Hum. Never realised I could just use "sexism" as a cover-all excuse for not doing my homework."
She later tweeted: "Seriously, though, sexism is still so rife and serious. This kind of thing trivialises it and makes it harder to fight."
Matt Chorley of the Times newspaper tweeted: "It would have been sexist if Murnaghan had asked Thornberry about Bake Off. Or her hairdresser. Not who she might talk to in government."
Leader of the Scottish Conservatives Ruth Davidson tweeted: "Plenty of genuine sexism & misogyny in politics. Don't need prominent women debasing the term to cover their own poor performance. Jeez."
But the Labour MP Paul Flynn came to Ms Thornberry's defence and tweeted: "Possibility of nuclear war is a issue of vital importance that should not be trivialised by Murnaghan's cheap smart-aleck pub-quiz question."
He later added: "Murnaghan repeats cheap trick to grab a headline for himself and his failing show by diverting attention from issue of mega-importance."
Ms Thornberry 'liked' a tweet sent by Pete Wishart, the SNP MP for Perth, which said: "Emily Thornberry quite right. Let's pub-quiz you on names instead of talking about the real issues."