You have a good memory, and so did that Witness. I had never heard of this argument before, but with some research I found an article in a 1985 Watchtower which explains this line of reasoning.
[Box on page 28]
How does First Chronicles help to prove that Jesus was not born on December 25?
According to First Chronicles, King David organized the priests into 24 “courses,” or groups, each group being assigned to serve for a week at the temple. Thus, a member of each course would get to serve at the temple twice each year, at approximately six-month intervals.
The first course began serving immediately after the end of the Festival of Booths, around late September/early October. The eighth group, named after Abijah, served a week in late November/early December, and then another week in late June/early July. Why is the division of Abijah significant? Because, according to Luke’s account, John the Baptizer’s father, Zechariah, belonged to “the division of Abijah,” and he was actually serving in the temple when the angel appeared to him and announced the coming birth of John.—Luke 1:5, 8, 9.
As Luke’s record shows, John was conceived very soon after this. Hence, he was born nine months later, either early September or early April. Luke’s record also shows that Jesus was six months younger than John. (Luke 1:26) Thus this detail from the book of Chronicles shows that, rather than being born at the end of December, Jesus was born either early March, or early October. Other scriptures show that the latter is the correct time.—For more details, see The Watchtower, June 15, 1954, page 382.
— w85 9/1 p. 28 "Read and Learn From the Two Books of Chronicles"
Nowadays, Jehovah's Witnesses commonly use the explanation that the shepherds would not have been out of doors at night (Luke 2:8) during the winter months, since winter in Bethlehem is especially cold and rainy.