Frenchman Quartararo, 21, posted a blistering one minute, 36.806 seconds on his satellite Yamaha bike to book a spot in the second qualifying session (1210 GMT), edging out Australian Jack Miller by 0.052 seconds.
Marquez, bidding for a seventh top-class title in the coronavirus-delayed season, was fourth-quickest, behind Suzuki’s Joan Mir and ahead of Yamaha rider Maverick Vinales.
But Honda’s Marquez left Suzuki’s Rins waving his arms in frustration as he tried to get past his compatriot.
Rins, who finished fourth in the championship last season, could only manage 12th and will have to contest the first qualifying session.
Italian veteran Valentino Rossi is through to second qualifying, although he finished 0.324 seconds slower than Quartararo despite riding a Yamaha factory bike.
The 2020 MotoGP world championship finally gets underway on Sunday, behind closed doors and more than four months late.
Fastest combined times from practice for the Spanish MotoGP at Jerez de la Frontera:
1. Fabio Quartararo (FRA/Yamaha-SRT) 1min 36.806sec, 2. Jack Miller (AUS/Ducati-Pramac) at 0.052sec, 3. Joan Mir (ESP/Suzuki) 0.102, 4. Marc Marquez (ESP/Honda) 0.223, 5. Maverick Vinales (ESP/Yamaha) 0.240, 6. Cal Crutchlow (GBR/Honda-LCR) 0.293, 7. Franco Morbidelli (ITA/Yamaha-SRT) 0.306, 8. Valentino Rossi (ITA/Yamaha) 0.324, 9. Francesco Bagnaia (ITA/Ducati-Pramac) 0.365, 10. Andrea Dovizioso (ITA/Ducati) 0.379