Not really, 13 as an unlucky number came long after 12 was used as the counting system in many cultures.

The reason 12 months was adopted is based on a confluence of things, but mainly the moon and your hands.

The moon: There's just over twelve lunar cycles per year, and the lunar cycle has been the longest used system to determine the advance of the seasons (for harvests and migration of animals). Month and Moon even have the same origin as a word.

Your hands: Because your hands are the origin of base 12 counting, which arose in many different civilizations as a way of doing simple math on your hands. This is why 12 is so prevalent in measurement. Time, Weight, Length, etc.

Notice how we all use our thumbs to count things out by touching finger tips, and we add in the thumb as number one? It is almost instinctual, but based on base 10 numbering.

Well that's great but with both hands it only gets you to five.

Now use your thumb, and count each of the spaces on your other four fingers between the wrinkles. You can get to 12 on one hand, 24 on two hands. Without a calculator or "advanced" knowledge of math, anyone can deal in higher numbers on the fly.