The 24-year-old right-hander is the son of Paul Quantrill, who pitched six of his 14 major league seasons with the Blue Jays.
The Quantrills are from nearby Port Hope, Ontario.
The rookie (0-2, 5.40 ERA) will face Blue Jays right-handed veteran Edwin Jackson (0-1, 6.30).
The Padres can clinch the three-game series with a win after taking the opener 6-3 Friday night.
The Toronto visit was circled on Quantrill's schedule even though he did not make the Padres' roster out of spring training.
"They decided I wasn't ready to start the season," Quantrill said. "But I said, 'I'm going to be there by the time we go to Toronto, and I'm not leaving.'"
He went 4-1 with a 4.80 ERA in six starts with Triple-A El Paso before he was called up on May 1. Quantrill was optioned to Triple-A on May 8 but recalled Sunday.
After allowing two runs in each of his first two major league starts, Quantrill allowed five runs over five innings Sunday at Pittsburgh and took the loss in the Pirates' 6-4 victory. He has yet to oppose the Blue Jays.
The first inning has been his problem. He has allowed a combined 11 hits in the first inning of his three major league starts, including five hits and four runs Sunday.
Opponents are batting .611 against him in the first inning and .233 after that.
"I think I've shown once I settle down that I dominate, I beat teams," Quantrill said. "I need to find a way to do that earlier. We're trying some different stuff. Some of it is I'm three starts into my career. It's hard to not be jacked out of your mind when you walk out there. That's not an excuse. I need it figure out how to fix that in the 'pen."
Jackson will make his third start for the Blue Jays on Saturday.
The 35-year-old last faced the Padres in 2017 and earned a win with seven innings of one-run ball. However, he is 1-6 with a 6.05 ERA in 16 career games (11 starts) against San Diego.
The Padres had another Canadian in their lineup Friday. Josh Naylor, from nearby Mississauga, Ontario, was 0-for-4 in his major league debut. He was called up earlier in the day after outfielder Alex Dickerson (sprained right wrist) went on the injured list.
The Blue Jays were busy with moves Friday that could have an impact Saturday. They promoted infielder Cavan Biggio, son of Hall of Famer Craig Biggio, from Triple-A Buffalo, and he was 0-for-3 in his major league debut.
Lourdes Gurriel Jr. also was recalled from Buffalo, and his first homer of the season tied the game 3-3 Friday.
Biggio, who played second base Friday, batted .307/.445/.504 with six home runs and 26 RBIs at Buffalo.
"I'm still trying to be the most consistent hitter I can be," Biggio said. "I worked on my direction a lot this offseason, and it's allowed me to put more balls in play."
Gurriel, previously a second baseman, played left field Friday. He had struggled both in the field and at the plate in 13 games with Toronto before being sent to Buffalo, where he worked on playing outfield so that he can be used in a utility role.
The Blue Jays also recalled left-hander Thomas Pannone from Buffalo and added left-hander Zac Rosscup, who was claimed off waivers from the Seattle Mariners on Thursday. Infielder Richard Urena and outfielder Billy McKinney were optioned to Buffalo.
Left-hander Ryan Feierabend was designated for assignment, and right-hander Elvis Luciano was placed on the bereavement list.
--Field Level Media