Quartararo, now the youngest ever back-to-back pole-sitter at 20 years old, had set the previous record in Saturday's final practice.
Suzuki's Spaniard Alex Rins completed the front row.
Marquez, on a factory Repsol Honda, will start off the front row for the first time this season. His team mate and fellow-Spaniard Jorge Lorenzo was ruled out for the weekend after crashing heavily on Friday.
Marquez has a 37 point lead over Ducati's Italian Andrea Dovizioso after seven of 19 races. Dovizioso qualified only 11th.
(Reporting by Alan Baldwin in London, editing by Christian Radnedge)
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