First let's define the seasons. Contrary to popular belief, what we think of as the seasons are actually the midpoints. Spring equinox is mid-spring, summer solstice is mid-summer (Shakespeare had this right!), fall equinox is mid-fall, and winter solstice is mid-winter. The cross-quarter days are very minor holidays in our modern times: Groundhog's Day is beginning of spring, May Day is beginning of summer, Aug 1 is beginning of fall, and Halloween is beginning of winter. I'm pretty anal about this so I use the astronomical cross-quarter days which are Feb 3/4, May 6/7, Aug 6/7, and Nov 6/7 depending on leap year. This year fall began on Aug 7 and will end on 11/7 (in the North Hemisphere).
I use this site from NASA to get the exact time of the seasons and cross-quarter days:
Next, head on over to one of my favorite sites, Time and Date. Look up the moon phases to find the last new moon before 11/7 which is 10/23. The moon is not visible in the sky during new moon. While you're here, take note of the 1st quarter which is 10/30; we'll come back to that.
From The Hobbit:
Soon, [Bilbo] saw the orange ball of the sun sinking towards the level of his eyes. He went to the opening and there pale and faint was a thin new moon above the rim of Earth...The sun sank lower and lower...The little moon was dipping to the horizon.
We need to find out when the moon is first visible after the new moon. This will vary from area to area. So we need a more universal way of setting this day. The best I can do is narrow it down. New moon is too soon and 1st quarter moon is too late. Split the difference and I come up with the day when the moon is exactly waxing crescent*, when the sun and moon is at a 45 deg angle. This puts Durin's Day at 10/27 this year. Purists may want to celebrate 2 days after the new moon on 10/25. At least this gives us a window.
*Since there are 8 points to the solar year, there should be 8 points to the lunar month.