The 19-year-old Swiss rider Dupasquier is in a “very serious condition” in hospital in Florence following a horrible accident during qualifying for the Moto3 race.
Quartararo admitted it was hard initially for him to regain his focus when the delayed FP4 session finally got underway following the accident, but would go on to claim pole with a new Mugello lap record of 1m45.187s.
“This morning when I make 1m45.6s and I saw Pecco [Bagnaia] was two tenths faster, I said ‘wow, in qualifying we can get close to 44 and 45 low’,” Quartararo said.
“And actually, the first run I did a mistake on my first lap and here I feel like it’s only the first lap of the tyre that’s good on our bike.
“But probably it’s one of my best laps off all time and I want to dedicate this one to Jason.
“I hope we will have great news coming.”
Ducati rider Bagnaia – who qualified second – said on Friday he expected at least seven or eight riders will be able to battle for victory in Sunday’s 23-lap Italian GP.
Quartararo is expecting a similarly chaotic race, but is confident Yamaha will be in the mix from the start now it has a front holeshot device.
“Yes, too many riders with good pace actually,” he said when asked about the competitiveness of the field after FP4’s race running.
“But Pecco, Johann [Zarco], [Alex] Rins, myself I think we have a really great pace and I’m quite happy about the front device that we have.
“At the beginning in FP1 I was struggling a little bit to use it, also strange in the first braking because you feel like your bike is super low.
“But on qualifying I made a really great start and step by step we will take confidence, and I feel like for tomorrow’s start we can do something great.
“Conditions will change. Here we know after Moto2 race it’s the one track where conditions change a lot.”
Quartararo also offered his thoughts on the qualifying tactics of Marc Marquez against his Yamaha teammate Maverick Vinales in Q1, believing the Honda rider’s actions were “borderline”.
“Was on the limit, honestly,” Quartararo said of Marquez’s tactics. “I had the same in Malaysia in 2019, but was not that much.
“For me I see it a little bit borderline because Maverick came into the pits but he [Marquez] came into the pits.
“This is not my job, I don’t want to judge anyone. But it was strange to see this image.”