April 6 comes from the date that the church (LDS) was originally organized. Talmage’s book has a chapter and a verse that states that Jesus was born on April 6, B.C 1. Talmage didn’t randomly select that time, he gets it from Section 20 of the D&C which states this,
“The rise of The Church of Christ in these last days, being one thousand eight hundred and thirty years since the coming of our Lord and Savior Jesus Christ in the flesh, it (the church) being regularly organized and established agreeable to the laws of our country, by the will and commandments of God, in the fourth month, and on the sixth day of the month which is called April.”
Through some interpretation and “praying” the majority of the church believes that this is Christ telling us exactly when he came into this world, at the precise date that the church was organized. Hope this helps.
Oh and December 5 is also a common one, people like J. Reuben Clark, a counselor in the first presidency at the time, and Bruce R. McConkie (need i clarify who he was?) all claim that Christ was born on December 5.
Cheers,
The Irishmen