The proverb was never about the colour. A green cheese is a fresh, unripe, pale one, and the joke was about a fool who mistakes a reflection in water for a round white cheese.

It is recorded from the sixteenth century as an example of an obviously absurd claim you could get a credulous person to swallow — so from the start it was a comment about gullibility rather than about the Moon. Nobody in the whole history of the phrase has ever believed it.