Open up the diplomacy screen with the civ you have an alliance with, and there should be a box at the bottom that says 'New' and 'Active'. Click on 'Active' and it should show you when your alliance will end.
I just count 20 turns from the time I sign the Alliance. It could end in 10 turns if you sign a Peace Treaty with the nation your against if they don't come to you first.
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