Carrier's hockey lineage is traced to his father, Bernard, and brother, Samuel, who each played for Quebec of the Quebec Major Junior Hockey League.

Samuel was selected by the Washington Capitals in the sixth round (No. 176) of the 2010 NHL Draft. Alexandre became a fourth-round pick (No. 115) of the Nashville Predators in the 2015 NHL Draft and as a rookie saw time on defense pairs with Roman Josi and Mattias Ekholm. Carrier was signed to a three-year contract April 15, 2020, an investment that has paid off for the Predators.

Carrier was on pace for the best season of his professional career before the American Hockey League suspended the 2019-20 season March 12, 2020, due to concerns surrounding the coronavirus. He scored 37 points (five goals, 32 assists) in 55 games for Milwaukee of the American Hockey League, the same totals he compiled in 76 games the season before.

Carrier scored his first NHL goal in the third period of a 6-3 loss at the Tampa Bay Lightning on March 13, 2021. He signed a three-year, $11.25 million to stay with the Predators on July 1, 2024, and was traded to the Montreal Canadiens for Justin Barron on Dec. 18, 2024.

He got his 100th point in a 6-3 win against the Vancouver Canucks at Bell Centre on Jan. 12, 2026. He scored twice in 20 seconds of the same game to become the first Canadiens player to do so since Cole Caufield (eight seconds; March 15, 2022). He also tied Bert Corbeau (20 seconds; Dec. 25, 1919) for the second-fastest two goals by a defenseman in team history, a list led by Larry Robinson (19 seconds; March 22, 1980), and was the first at his position to score two goals within 20 seconds since Dustin Byfuglien (16 seconds; Nov. 30, 2007).