Conception date would give you fetal age, not gestational age (with the typical female cycle ovulation occurs on Day 14 after first day of LMP. If you conceive on that day, your baby's gestational age is 2 weeks the day it is conceived but fetal age is 0). Doctors use gestational age/lmp because most women do not know when they ovulated but many know when their lmp was.
That said a due date of Sept 23; subtract 40 weeks you get a presumed lmp of Dec 17. Add 14 days your conception date was around Dec 31.
Well when I was pregnant, I was due Sept 17th, 2010 and my conception day was something like Dec. 24-25th. Yours would be sometime in early Jan. More specifically, December 28 to January 5, 2011. This however is ONLY a guess.
Conception date however isn't actually important unless paternity is called into question. If this is the case, you need DNA testing done once the baby is born. It wouldn't be something you would guess about.