In the morning's opening session, Quartararo, on a Yamaha, recorded the fastest time and was the only rider to lap under 1 minute and 32 seconds.
In the afternoon, as the weather grew overcast the laps grew quicker and Vinales, on a factory Yamaha, and championship leader Marquez on a Honda beat the young Frenchman's earlier time.
"It's been a very positive start to the weekend for me. I think we found our set-up very fast, so I was able to make many laps and I felt quite good with the bike," said Vinales. "Let's see what the weather will be like, because it seems to be uncertain."
Quartararo, who races for Yamaha's Petronas development team, said the course seems to suit the machine.
"I think this track is good for the Yamaha," he said. "We saw Vinales go really fast today."
Marquez nearly came off his bike in the first session.
"I made a big save in FP1, it was my second run and I was able to save it but it took me by surprise a little bit because it's so bumpy," Marquez said. "Another nice video for the collection!"
The other Honda rider Jorge Lorenzo did crash, adding to the impression that the bikes are a little unstable.
"Today didn't start in the ideal way with the crash this morning," Lorenzo said. "I was running with too much lean over the curbs and wasn't able to save it."
Alex Rins, one point behind compatriot Marquez in the standings, was only 16th on his Suzuki.
"This track always produces tight lap times with many riders running very close together," the Spaniard said. "I'm down in 16th place, I'm just a 10th of a second from 10th."
Valentino Rossi, who is fourth in the standings, was 14th after snapping the chain on his Yamaha in the opening session.
"Sincerely, I expected to be stronger, but unfortunately I wasn't very fast," he said.
Quartararo said he was enjoying his home GP.
"I don't feel like I'm under pressure and I'm having fun with the fans," the Frenchman said.
Rain is forecast for both Saturday, when free practice is at 1230 GMT and qualifying starts at 1600, and for the race on Sunday.
Combined times after Friday's two free practice sessions:
1. Maverick Vinales (ESP/Yamaha) 1min 31.428sec, 2. Marc Marquez (ESP/Honda) at 0.190, 3. Fabio Quartararo (FRA/Yamaha-SRT) 0.285, 4. Jorge Lorenzo (ESP/Honda) 0.288, 5. Andrea Doviziosio (ITA/Ducati) 0.308, 6. Danilo Petrucci (ITA/Ducati) 0.471, 7. Pol Espargaro (ESP/KTM) 0.495, 8. Aleix Espargaro (ESP/Aprilia) 0.704, 9. Takaaki Nagakami (JAP/Honda-LCR) 0.775, 10. Jack Miller (AUS/Ducati-Pramac) 0.810