How many times have the Winnipeg jets won the Stanley cup?

Zero. But they did win the World Hockey Association's championship trophy, the Avco Cup, in 1976, 1978, and 1979. In 1996 they relocated to Phoenix and were renamed the Coyotes.

In 2011, the Atlanta Thrashers moved to Winnipeg and became the Jets.

The Winnipeg Jets have never won the Stanley Cup. The only Winnipeg team to win the Cup were the Winnipeg Victorias who did it twice each in 1896, 1901 and 1902. The reason they are on the Cup twice for each of those years is because the Stanley Cup was originally a Challenge Cup, meaning that in addition to defending the Cup through some sort of playoff format at the end of the season, the defending Champion usually had to defend it against a team who had formally filed a challenge with the trustees of the Cup at some point during the season as well.