No. It is not explained in the Marvel Cinematic Universe ever except the conversation you already mentioned. This may or may not be explained in future films, but it is pure speculation for now.
However, it is explained in the comic world.
In Earth-616, Nick Fury fought in World War II as part of the Howling Commandos. A grenade by Nazi soldier was thrown at him. Though he caught it, but delayed throwing it back. The shrapnel he was hit with was what caused the damage that would slowly deteriorate the vision in his left eye, eventually leaving him almost completely blind in that eye.
In the Ultimate Marvel Universe, his eye was injured due to a bombing he was caught in alongside James Howlett — a.k.a. Wolverine/Logan — during the Gulf War, but unlike his Earth-616 counterpart, he completely lost the eyeball and was left with bad scarring around the now empty socket.
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