Joseph was selected by the Saint John Sea Dogs in the 2013 Quebec Major Junior Hockey Leagueentry draft in the third round, 51st overall. He began to play for the Sea Dogs at the midway point of the 2013–14 season. His first goal with the team came in a 2–0 win over the Val-d'Or Foreurs. In his second season with the Sea Dogs, Joseph became the sixth player in team history to score five points in a regular season game. He also finished fourth in team scoring that season with 42 points. On June 27, 2015, the Tampa Bay Lightning selected Joseph in the fourth round, 120th overall, of the 2015 NHL Entry Draft. On December 24, 2016, the Lightning signed Joseph to a three-year, entry-level contract. At the time of his signing, Joseph had played in 176 career games with the Sea Dogs during which he scored 80 goals and 171 points. During the 2016–17 playoffs, Joseph was named Canadian Hockey LeaguePlayer of the Week after scoring two goals and three assists in two games. Joseph went on to help the Sea Dogs capture the President's Cup as QMJHL champion over the Blainville-Boisbriand Armada in a four-game sweep. The Sea Dogs went on to lose in the semifinals to the Ontario Hockey League's Erie Otters in the 2017 Memorial Cup.
Professional
Joseph began his professional career in the 2017–18 season with the Tampa Bay Lightning's American Hockey League affiliate, the Syracuse Crunch. He had a strong rookie season with the Crunch, including being named CCM/AHL Rookie of the Month for March 2018, during which he scored 4 goals and 10 assists for 15 points over 12 games; he also had a seven-game scoring streak within that span. Joseph finished the season with 15 goals and 38 assists for 53 points, leading the Crunch in regular season assists and points. Joseph also had three goals and seven points in seven Calder Cup playoff games. At the conclusion of the Lightning's training camp ahead of the 2018–19 season, Joseph found out he had made the opening night roster from general manager Julien BriseBois—before boarding the team's charter plane after a 3–2 Lightning win against the Florida Panthers, BriseBois informed Joseph he had made the roster. On October 6, 2018, Joseph made his NHL debut in a 2–1 Lightning win against the Florida Panthers at Amalie Arena. On October 16, he scored his first career NHL assist and point in a 4–2 Lightning win over the visiting Carolina Hurricanes. On November 4, Joseph scored his first NHL goal in a 4–3 Lightning overtime win over the Ottawa Senators away at the Canadian Tire Centre. Joseph was one of the eight players called up to the Lightning for their training camp prior to the 2020 Stanley Cup Playoffs.
International play
On April 29, 2019, Joseph was named to make his senior debut for Team Canada at the 2019 IIHF World Championship held in Slovakia. He helped Canada progress through to the playoff rounds before losing the final to Finland to finish with the Silver Medal on May 26, 2019. Joseph finished the tournament posting 1 goal and 2 points in 10 games.
