There’s always been some conflict about the date of Christ’s birth. In fact, as a Mormon I thought He was born in the spring, on April 6, in the year 4 or 5 BC. New research points to the birth of Christ sometime in December of 5 BC. That would certainly help the spirit of Christmas.
BYU professor, Jeffrey R. Chadwick, Jerusalem Center Professor of Archeology and Near Eastern Studies, published his research in BYU Studies, “Dating the Birth of Jesus Christ” which definitely challenges the April 6 date.
Looking at all the research, and documents available — scriptural, historical, archeological, and astronomical– Chadwick narrows it down to December of 5 BC as the probable date of the Savior.

Why do Mormons think Christ was born on April 6th?

Chadwick is careful to explain that the Doctrine and Covenants section 20 was a heading written by scribe, John Whitmer on April 10. The manner of writing out the year, (one thousand eight hundred and thirty years since the coming of our Lord and Savior Jesus Christ in the flesh) was typical for that time and was a recording of the organization of the church — not a revelation on the birth of Christ.
*The Church has taken no official position of the exact date, even though some authorities have referred to Christ’s birth as being on the same day as the organization of the church. New evidence in the Joseph Smith Papers has made this clear.
For more detail you can read  here.  And an article that supports the April birth can be found here.

Artwork of the Birth of Christ: