I will add that they do one of many things: The skin will wrinkle, the neck will "droop", the skin will crack, the fig will fall off, the fig will become soft enough to easily "squish" it between your fingers, and/or it will ooze from the eye (could be any or all of the above).
My BT typically will begin to start turning blushing reddish-purple on any part that is in constant sun about 2-3 weeks prior to the figs actually getting ripe like THIS and THIS - When it is about 7-10 days from ripe the skin looks like THIS - When it is fully ripe, it looks like THIS and THIS although I will tell you - that one was fairly overripe (it's the only way BT figs have ANY flavor), normally a "nicely ripe" BT will have yellowish green around the very top of the fruit.
I have several breba figs on mine now, and they're still about 4-5 weeks from being fully ripe, they are still very firm, very green (with white flecks) and the neck is so stiff that some are sitting out horizontal from the ground.
If you bought your tree online at Willis Orchards or TyTy, or you bought it at a hardware store like Home Depot or Lowes, you have a great likelihood it was mislabelled. Don't buy fruit at those kinds of stores unless you don't care what variety it is.
Post pictures of the fruit you are seeing. That will help people to tell you if it's ripe. A picture is always worth a thousand words. You can type explaining your fruit till your fingers fall off ... and still not give a good visual. Or you can post 2-3 pictures and all confusion is gone.