A native of Denver, Terry played with the Colorado Thunderbirds under-16 team at the Tier 1 Elite Hockey League from 2008 until 2014.

He earned various nicknames throughout his mini mite career including "Coach Troy" and "Two and a Half Men", due to his size.

Due to his accelerated high school career, Terry became one of college hockey’s youngest players as a freshman.

[9] During his freshman and sophomore season at Denver, Terry worked with the Pioneer's head strength and conditioning coach to be more of an impactful player on the ice.

[10] During his freshman season, the Pioneers made it to the Frozen Four of the NCAA Tournament but fell to the eventual champions, North Dakota.

[14] Terry was immediately added to the Ducks' roster for their postseason push, and made his NHL debut the same day in a 4–1 loss to the Vancouver Canucks at Rogers Arena.

In his debut, Terry skated on the Ducks' third line with Adam Henrique and Ondrej Kase and registered two shots on goal in 12:30 minutes of ice time.

[23] On July 14, 2020, as a restricted free agent, Terry was signed by the Anaheim Ducks to a three-year, $4.35 million contract extension.

[22] When the NHL returned for the 2020–21 season, Terry remained on the Ducks' roster prior to their opening night game.

[33] The following year, Terry helped Team USA compete for a gold medal at the 2015 IIHF World U18 Championships after scoring 3 goals on 3 attempts in the shootout to eliminate Russia in the tournament.

[35] As the NHL refused to allow their players to compete at the 2018 Winter Olympics, Terry and other collegiate athletes were chosen to play for the United States men's ice hockey team.