From Quora :
Yes. It's pretty much the same word. In Old English:
mona = moon
monath = month (spelled monað)
The Oxford English Dictionary says that they're basically the same word with the same etymology, just with a slightly different ending.
did the word month come the word moon?
Yes. Since the lunar month always began with the appearing of the new moon (Heb., cho′dhesh), the term “new moon” also came to mean “month.”