Does Doctrine and Covenants 20:1 suggest that April 6 should be regarded as the exact date of the Savior’s birth? The article "Dating the Birth of Jesus Christ" says probably not.
Elder James E. Talmage, in the book Jesus the Christ, was apparently the first LDS writer to propose that Jesus was born on April 6 in the year 1 BC.
However, today, virtually all hard evidence, scriptural and otherwise, points to a narrow window of time at the beginning of winter for the birth of the Savior, and that Jesus was quite likely born in December of the year 5 BC.
Latter-day prophets from Joseph Smith to Lorenzo Snow made no specific comments on the date of Jesus’s birth.
During the 20th century, 3 different LDS Apostles published major studies on the life and ministry of Jesus Christ and in them offered models for the date of Jesus’s birth:
1. Elder James E. Talmage (see above).
2. President J. Reuben Clark, of the First Presidency, published Our Lord of the Gospels in 1954, which was used as the Melchizedek Priesthood manual in 1958. In it, he pointed to the traditional early winter time frame for the date of Jesus’s birth. He explained: “I am not proposing any date as the true date. But in order to be as helpful to students as I could, I have taken as the date of the Savior’s birth the date now accepted by many scholars,—late 5 b.c. or early 4 b.c. In the timetables he employed in his book, Clark listed his preferred time range for Jesus’s nativity as December of 5 bc, and the time range of the Annunciation to Mary as nine months earlier in March of 5 bc.
3. Elder Bruce R. McConkie, in The Mortal Messiah, reviewed the positions and reasoning of both Talmage and Clark with regard to Jesus’s birth date and stated that he would follow Clark’s course. Accordingly, McConkie dated the Annunciation to Mary in March or April of 5 bc, and the birth of Jesus in December of 5 bc (with the caveat that his birth could also have occurred from January to April of 4 bc).
It seems there is no authoritative agreement or position on the issue of the birth date of Christ that can be regarded as authoritative.
Also read Ronald Millett's article "Elder James E. Talmage and the Birth Date of Jesus Christ."