I'm trying to say very briefly that the night was starry and there was the moon also.
Is it correct to say moon starry night? If not, do you see any option?
Maybe mooned starry nights?
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Normally if it is a moonlit night, the stars are not as bright as they are on a night without a moon, but if you mean to say that the night has both a moon and bright stars, you could say that many ways:
"The sky was lit by the moon and the stars."
"The night was starry and moonlit."
"It was a starry, moonlit night."
But "moon starry night" does not sound idiomatic at all to me.
It's fine to say starry night - here, starry is a adjective applied to the noun night.
We can't say
moon night - we need to turn moon into a adjective. One guess might be moony, but if we look that up, it has very different meaning. The word we're looking for is moonlit, which is an adjective we could use here:
It was a moonlit starry night.
Other answerers have already pointed out that
It was a starry moonlit night.
fulfills your request adequately. However, word choice can also depend on context — is this supposed to sound poetic? archaic? matter-of-fact?
The night was lit by the moon and stars above.
The only source of light was the moon and stars.
The stars, almost as bright as the full moon, cast a mystical light over the scene below.
Deneb and Aldebaran blazed cheerily as always, but for once there was only a single moon in the sky.
On a night when both the moon and the stars are visible, set up your telescope on the highest point you can find.
In other words, it's hardly necessary to state that the night was lit by the moon and stars; those are by definition the things that are out at night. We don't see many sunlit nights! (At least not at my latitude.) So if you're explicitly mentioning the moon and stars, there must be a reason for it. Find that reason and work it in. With more creative possibilities, you'll have correspondingly less need to worry about one particular construction sounding stilted.
Also notice that while the moon comes and goes, the stars are always "out" — unless it's cloudy. Thus another option:
The night was moonlit and clear.