Actually your location indicated in the upper right above your minimap will have a number (i.e. Vulcan Sector #1) this is your instance number. If the name is too long you can hover over it with your mouse and it should show the whole name including the number.
This would be a great idea! Glad to see I'm not the only one who is bothered by this omission. Even better, would be for Fleet members to be prompted to join the same instance AND stay in the same instance. STO is the worst MMO I've seen for joining up with friends in game.