Cote, a native of Quebec City whose younger brother Alain also played in the NHL, was selected by the Hartford Whalers in the first round (No. 11) of the 1984 NHL Draft. However, the defenseman had his best offensive seasons with the Washington Capitals in the early 1990s.
The Whalers chose Cote after he scored 15 goals and 65 points in his second season with Quebec of the Quebec Major Junior Hockey League. He spent almost all of his age-18 season with the Whalers, finishing with three goals and 12 points in 67 games. But he played just two games with Hartford in 1985-86, splitting most of the season between Hull of the QMJHL (where he was the league's best defenseman despite playing just 26 games) and Binghamton of the American Hockey League.