The relation between these is a bit tenuous. The real cause of the Gulf War was the US's need to ensure the safety and relative stability of the major oil fields in places like Kuwait and Saudi Arabia. The Soviet Union (then in its last days) did not block UN resolutions allowing the war to happen. In other words, the war itself was not a Cold War conflict.
The connection between the two goes back farther. It is connected to the Iranian Revolution of 1979 and the ensuing Iran-Iraq War. When it became clear that the anti-US Iranian regime could win the war, the USSR started helping Iran and the US started helping Iraq. US help to Iraq helped to build Saddam Hussein's military, which gave him the ability to invade Kuwait, thus causing the Gulf War. So there is a connection, but it is not a very direct connection.