The 20-year-old Quartararo has emerged as a fascinating subplot this season as he bids to end a 20-year French barren run in MotoGP by winning his first race after five pole positions and six podium finishes in his debut top-flight season.
On the 4km Ricardo Tormo circuit, the rider edged out fellow Yamaha man Maverick Vinales in the second session, with six-time top-class world champion Marquez coming third.
Former three-time MotoGP world champion Jorge Lorenzo, 32, was way down the rankings in 16th after announcing he will retire after the race.
Italian veteran Valentino Rossi fell twice on Friday, going down in a shower of sparks as his bike hit the ground. He finished the day 14th.
Marquez, who was crowned MotoGP champion for a sixth time last month, is looking for only his second MotoGP win in Valencia and was just 0.077 seconds slower than Quartararo.
Marquez has undoubtedly been the star of the 2019 season, winning 11 races and wrapping up the title with four races to spare.
No one has come close to the 26-year-old's speed and consistency but he has only won once in MotoGP at Valencia, back in 2014.
Leading combined MotoGP times:
1. Fabio Quartararo (FRA/Yamaha-SRT) 1min 30.735sec, 2. Maverick Vinales (ESP/Yamaha) at 0.148, 3. Marc Marquez (ESP/Honda) 0.239, 4. Jack Miller (AUS/Ducati-Pramac) 0.265, 5. Franco Morbidelli (ITA/Yamaha-SRT) 0.464, 6. Alex Rins (ESP/Suzuki) 0.495, 7. Joan Mir (ESP/Suzuki) 0.545, 8. Aleix Espargaro (ESP/Aprilia) 0.570, 9. Andrea Dovizioso (ITA/Ducati) 0.616, 10. Johann Zarco (FRA/LCR-Honda) 0.634
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