Of the two, Honda rider Marquez was the first to take the lead early, but Quartararo quickly demoted him with the first time of less than two minutes of the session.
Two improvements from Marquez saw him recapture first place by less than a tenth, five minutes from time.
But Petronas SRT rider Quartararo could get faster too, clocking 1: 59.609 minutes in the final minute of the warm-up, beating Marquez by 0.07 seconds.
Ducati’s Andrea Dovizioso was the best of the others, three tenths slower than Quartararo. Front row Valentino Rossi (Yamaha) moved up to fourth on a last minute lap after spending most of the session outside the top 10.
Suzuki’s Alex Rins finished fifth while Maverick Vinales took a third Yamaha into the top six.
Local hero Cal Crutchlow’s progress in the session was halted with an early pit stop, but the LCR Honda rider recovered in the top 10, finishing seventh from Quartararo’s team-mate Franco Morbidelli.
The Ducati duo Danilo Petrucci and Jack Miller (Pramac) completed the top 10, with the latter almost falling at Stowe and having to go through the gravel.
Honda rider Jorge Lorenzo was the last, 3.474 seconds behind the benchmark time.