The Brazilian's last-eight opponent will be world No. 3 Alexander Zverev who dispatched Zizou Bergs 6-2, 7-5.
Fonseca, who is making his first appearance in Monte Carlo, produced some superb tennis to see off an opponent who had thrashed Daniil Medvedev 6-0, 6-0 in the second round.
The teenager is the youngest man to reach this stage in Monte Carlo since Rafael Nadal and Richard Gasquet in 2005, and is the first Brazilian to reach a Masters 1000 quarter-final since Thomaz Bellucci in Madrid in 2011.
He is the second man born in 2006 to reach the quarter-finals at a 1000-level event, after Spaniard Martin Landaluce, who reached the last eight in Miami in March.
"It is super special. I was looking for this result for a long time," Fonseca said on court.
"Of course I want more. I am very confident and focused. I was very happy with the way I fought today. From the beginning, putting a lot of pressure. Playing huge return games and very good serve games. I was putting a lot of pressure and that helps you stay more calm during the match."
The world number 40, who is up five places in the live rankings, took just 73 minutes in the Monaco sun to book his place in the last eight, stamping his ticket with a winning smash on his first match point.
After dispatching Bergs, Zverev praised Fonseca.
"I think clay is his best surface," Zverev said of the Brazilian. "I am excited to play him for the first time. He is a young upcoming talent and I think we will play each other a lot more in the next couple of years."
Zverev, a two-time semi-finalist at the tournament, ruled the opening set against Bergs as the Belgian struggled on his own service, holding only once.
Bergs found his serve in the second set, but Zverev managed to break at 3-3.
The German lost his rhythm when he was serving for the match at 5-4. Bergs swinging freely broke to love after committing four unforced errors.
Zverev responded immediately by breaking back. He closed out the match in his next service game with an ace on his second match point to win in 1 hour and 51 minutes.
"(It was) a better match from my side," Zverev said when asked how it compared to his opening-round win on Wednesday in Monte-Carlo, where he rallied from 2-5 in the deciding set to defeat Cristian Garin.
"It was still not perfect. Still far from the level I played in the US, but this is my first clay tournament... I am happy with the level today. We move forward."
The No. 3 player in the ATP rankings, Zverev, is chasing his eighth Masters 1000 crown and will now meet Fonseca for the first time. Last month, Fonseca faced Carlos Alcaraz and Jannik Sinner for the first time in Miami and Indian Wells, pushing both in narrow defeats.