Towards the end of [his April 2014 General Conference address, Elder Bednar stated (emphasis added):

Today is April 6. We know by revelation that today is the actual and accurate date of the Savior’s birth. April 6 also is the day on which The Church of Jesus Christ of Latter-day Saints was organized. On this special and sacred Sabbath day, I declare my witness that Jesus the Christ is our Redeemer. He lives and will cleanse, heal, guide, protect, and strengthen us. Of these things I joyfully testify in the sacred name of Jesus Christ, amen.

His talk would be published in the Ensign with this parenthetical note:

See D&C 20:1; Harold B. Lee, “Strengthen the Stakes of Zion,” Ensign, July 1973, 2; Spencer W. Kimball, “Why Call Me Lord, Lord, and Do Not the Things Which I Say?” Ensign, May 1975, 4; Spencer W. Kimball, “Remarks and Dedication of the Fayette, New York, Buildings,” Ensign, May 1980, 54; Discourses of President Gordon B. Hinckley, Volume 1: 1995–1999 [2005], 409.

April 6th birthday references seem to depend primarily upon James Talmage’s analysis of D&C 20:1 in Jesus the Christ (emphasis added):

As to the season of the year in which Christ was born, there is among the learned as great a diversity of opinion as that relating to the year itself. It is claimed by many Biblical scholars that December 25th, the day celebrated in Christendom as Christmas, cannot be the correct date. We believe April 6th to be the birthday of Jesus Christ as indicated in a revelation of the present dispensation already cited [D&C 20:1], in which that day is made without qualification the completion of the one thousand eight hundred and thirtieth year since the coming of the Lord in the flesh. This acceptance is admittedly based on faith in modern revelation, and in no wise is set forth as the result of chronological research or analysis. We believe that Jesus Christ was born in Bethlehem of Judea, April 6, B.C. 1.

Harold B. Lee explicitly references D&C 20:

April 6, 1973, is a particularly significant date because it commemorates not only the anniversary of the organization of The Church of Jesus Christ of Latter-day Saints in this dispensation, but also the anniversary of the birth of the Savior, our Lord and Master, Jesus Christ. Joseph Smith wrote this, preceding a revelation given at that same date [he then quotes D&C 20:1].

Kimball merely refers to his birthday on April 6th (presumably also at 1 BC) as a fact.1

Hinckley uses slightly stronger language but never indicates a scriptural reference or even a recipient of the revelation (emphasis added):

[I]n ancient times Christmas, commemorating the birth of the Christ child, was celebrated at this solstice season [on December 25]. Men had no knowledge of the time of His birth, and so they came to bond the celebration of Christmas with the celebration of the return of the sun.

While we now know through revelation the time of the Savior’s birth, we observe the 25th of December with the rest of the Christian world.

The Book of Commandments and Revelations manuscript of D&C 20 was recently discovered, and it seems to undermine the claim that D&C 20 was meant, at the time of its writing, to be indicative of Jesus’s birthday. Deseret News spoke with Steven Harper, a BYU Assistant Professor of Church History and volume editor of the Joseph Smith Papers, about the significance of this discovery on the meaning of D&C 20:

… some people, including Elder Talmage, have read this verse as if it is the Lord speaking and revealing precisely that Christ was born 1,830 years before that day and that the revelation was given on April 6, 1830.

The recent discovery of the Book of Commandments and Revelations manuscript of D&C 20, however, showed that the verse was actually an introductory head note written by early church historian and scribe John Whitmer — something he did for many of the revelations, Harper said. “So those are separate from the texts that Joseph produces by revelation.”

The manuscript, published as part of the Joseph Smith Papers, also shows that the revelation was given on April 10 — not April 6. So although it references the organization of the church a few days earlier, the revelation — which topically has nothing to do with the birth date of Christ — and its introductory verses “shouldn’t be read as if it is a revelation of the birth date of Jesus Christ,” Harper said. “The interpretation that has been most popular over time is very much subject to question; that’s all I’m saying.”

And this wasn’t the only time that John Whitmer would identify a date with similar language. Another time he wrote, “It is now June the twelfth, one thousand eight hundred and thirty one years, since the coming of our Lord and Savior in the flesh.”

In other words, this type of language was merely a fancy 19th-century way of saying the date.

[unfinished from here on; need to complete with the below material]

https://byustudies.byu.edu/content/volume-1-chapter-24

On the 6th of April, in the land of Zion, about eighty officials, together with some unofficial members of the Church, met for instruction and the service of God, at the Ferry on Big Blue river near the western limits of Jackson county, which is the boundary line of the state of Missouri and also of the United States. It was an early spring, and the leaves and blossoms enlivened and gratified the soul of man like a glimpse of Paradise. The day was spent in a very agreeable manner, in giving and receiving knowledge which appertained to this last kingdom—it being just 1800 years since the Savior laid down His life that men might have everlasting life, and only three years since the Church had come out of the wilderness, preparatory for the last dispensation. The Saints had great reason to rejoice: they thought upon the time when this world came into existence, and the morning stars sang together, and all the sons of God shouted for joy; they thought of the time when Israel ate the “Passover,” as wailing came up for the loss of the firstborn of Egypt; they felt like the shepherds who watched their flocks by night, when the angelic choir sweetly sang the electrifying strain, “Peace on earth, good will to man;” and the solemnities of eternity rested upon them. This was the first attempt made by the Church to celebrate the anniversary of her birthday, and those who professed not our faith talked about it as a strange thing.

  1. Spencer W. Kimball, in his 1975 address, merely mentions the “fact” of April 6th in passing:

    The name Jesus Christ and what it represents has been plowed deep into the history of the world, never to be uprooted. Christ was born on the sixth of April. …

    A passing statement in Kimball’s dedicatory remarks:

    today we not only celebrate the Sesquicentennial of the organization of the Church, but also the greatest event in human history since the birth of Christ on this day 1,980 years ago.

    ↩