Fabbro was originally drafted by the Seattle Thunderbirds in the 1st round (8th overall) of the 2013 WHL Bantam Draft, but decided to play with the Penticton Vees of the BCHL to retain his NCAA eligibility. During the 2015–16 BCHL season, Fabbro was named an alternate captain and ended the season being named the Best Defenseman in the league. He committed to play for Boston University for the 2016–17 season before being drafted 17th overall in the 2016 NHL Entry Draft. During his sophomore season, Fabbro was selected for the Hockey East Second All-Star Team after coming in second among Hockey East defensemen in points.

On March 27, 2019, Fabbro signed a three-year, entry-level contract with the Nashville Predators. He made his NHL debut a few days later on March 30, in a 5–2 loss to the Columbus Blue Jackets and recorded his first career NHL goal on April 9, 2019.

On April 29, 2019, following his first playoff experience with the Predators, Fabbro was named to the Team Canada roster for the 2019 IIHF World Championship held in Slovakia. Fabbro helped Canada progress through to the playoff rounds before losing the final to Finland to finish with the Silver Medal on May 26, 2019. He finished the tournament posting 1 goal and 3 points from the blueline in 9 games.