Hague signed a three-year entry-level contract with the Golden Knights on September 29, 2017, and attended their training camp before rejoining the Steelheads for the 2017–18 season.[3] Following a 2017–18 season where Hague led his team in scoring and set a Steelhead single season record for defencemen (78 points), Hague was awarded the Max Kaminsky Trophy, which is awarded to the OHL's Most Outstanding Defenceman of the Year.[4] On April 8, following the Steelheads elimination from the 2018 OHL Playoffs, the Golden Knights assigned him to their American Hockey League (AHL) affiliate, the Chicago Wolves.[5]
With the Golden Knights becoming known for aggressively trading draft picks and drafted players, Hague was notably the only player drafted by the organization to play in the 2023 Stanley Cup Finals.[7] Hague scored a goal in the series-clinching Game 5 against the Florida Panthers to help the Golden Knights win the Stanley Cup.[8]