The shadow foreign secretary clashed with Mr Murnaghan as she criticised him for “pub quizzing” her.
Ms Thornberry was asked if she had taken part in any Brexit talks with her would-be counterparts in Germany and France.
She was then asked if she knew the name of the French foreign minister, Jean-Marc Ayrault.
“Don’t start pub quizzing me, Dermot,” she said.
She was then asked again if she was able to name the minister.
She said: “No and I’m not going to start answering your questions on this.”
She continued: “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.
She added: “You really want to spend this time pub quizzing me? What’s with you?”
She was later asked if she knew who the South Korean president is.
An increasingly frustrated Ms Thornberry said: “If you want me to go with you to a pub tonight and we can do a pub quiz tonight, let’s do it.”
Then asked if she knew the president’s gender, she said: “I’m not getting drawn by you into this nonsense.”
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 continued: “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.
“I really do. 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 that 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 told the presenter 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’.