[3] In his second season with Brandon, and his first full one in the WHL in 2018–19, Greig recorded 14 goals and 21 assists for 35 points in 63 games.

[11] Greig made his NHL debut on January 25, 2023, in Ottawa against the New York Islanders playing on a line with Alex DeBrincat and Claude Giroux.

[16] Early in the season, he missed ten games after suffering a high ankle sprain, returning December 4, playing on the third line with Dominik Kubalík and Mathieu Joseph.

[18] On February 10, in a 5–3 victory over the Toronto Maple Leafs, Greig used a slapshot to score a goal on an empty net very late in the game, a play Maple Leafs' defenceman Morgan Rielly took issue with and skated over and cross-checked Grieg in the face.

[21] During the Senators training camp ahead of the 2024–25 season, Greig suffered an upper-body injury in an exhibition game with the Toronto Maple Leafs.

[22] Then in mid October, he suffered another upper-body injury,[23] forcing him to miss a week of games.

[25][26] Greig was selected for Team Canada's roster for the 2022 World Junior Ice Hockey Championships.

[31] Greig accepted an invitation to make his senior national team debut at the 2024 IIHF World Championship.

[33] Greig's father Mark played nine seasons in the National Hockey League (NHL) for the Hartford Whalers, Toronto Maple Leafs, Calgary Flames and Philadelphia Flyers.