I found a few quotes. The phrase has been around for awhile, but it seems to have picked up in the 60's. In particular, the October 1968 conference had that phrase mentioned more than the previous two decades combined. My guess is that the phrase picked up steam and was promoted by the right wing cohort of Lee, Smith, McConkie, and Peterson. These quotes don't come specifically from those folks, but it's consistent with the ideology.
It's a very fundamentalist claim, that not only are the church's claims literally fact, but they are exclusively true. This is echo in the D&C which claims the church to be the "only true and living church on the face of the whole earth". However, I think there's a distinct difference between the way that people like McKay or Hugh Brown understood this phrase and the way that the right wingers understood it. For McKay, "true" was more akin to the idea that God is generally leading the church in the correct direction, not that it was always exactly where God wanted it to be. In other words, true didn't mean that the church wouldn't drift from time to time. For people like McConkie, "true" meant that the church's authority in the priesthood was absolute. If you look at what was happening at the end of the 60's and beginning of the 70's, the church was terrified of the growing radical youth movement and drug culture. They were freaking out over new Mormon scholarship and reacted by instituting correlation and centralizing power. For them, emphasizing "the church is true" was equivalent to a loyalty oath. Obedience became the key word. If the church is true, then you have to be obedient to it. There was no middle ground.
So I could easily see this as the starting point of emphasis for this now common testimony phrase.
Anthon H. Lund, Conference Report, April 1904, p.76 wrote:This statement which has just been read in your hearing was made for the Church. Rumors have been afloat that plural marriages have taken place, and some are said to have commenced to doubt the truth of the declaration made by our President at Washington. Now it has been laid before you, and the Church, by its vote in solemn assembly, has ratified this resolution, and the Saints know just where the Church stands on this question. If any come to you with such rumors, you know that the Church is true to that which it accepted thirteen years and six months ago, and which it has again ratified here in this Conference.
John G. Allred, Conference Report, April 1927, p.119 wrote:
God has chosen pastors after his own heart, and he has planted in the hearts of those pastors an individual testimony that God lives. Is it not true all over the missions? Ask these mission presidents, if the young men and women who come to them, many times without this testimony, do not in a few months time stand up and say, "I know the Church is true." That is the thing the world needs today--men and women who know whereof they speak, and who are able to back up what they say by the prophecies and teachings of the prophets of God.
Rey L. Pratt, Conference Report, April 1930, p.130 wrote:
I feel just as sure as I feel that I live that this Gospel will thrive. It doesn't worry me what men think about whether we are going to succeed or not. I know that there are people in the Church that shouldn't be in the Church. I have met a lot of people that to me seem to be just like the foam that the waves beat up upon the shore and that then melts away. There is also a stratum that is in some parts and factions of the Church that is like the dregs of poison. But the vast body of the Church is true and is clear and is pure, and this work will go on to ultimate success and will remain. It is, and always will be, the power of God unto salvation.
Joseph J. Daynes,Conference Report, October 1936, p.59 wrote: And I testify to you that I know that this Church is true, that the Gospel of Jesus Christ and the Church of Jesus Christ of Latter-day Saints are one and the same, inseparable and indivisible. As I beauty and its glory, and in my work I am happy that I am one who is undertaking to promote its glorious truths.
S. Dilworth Young, Conference Report, April 1948, p.103 wrote:
I stood and talked on the restoration of the Book of Mormon for twenty minutes or so, the most inspired, easiest talk I ever gave. It was promoted by the Holy Ghost; I know it was. I bore my testimony humbly and fervently to the truthfulness of the Book of Mormon, their book, and that the Church is true, and that Jesus lives. I told them of the vision of Joseph Smith, of the bringing forth of the book, about the Witnesses, the need for the Book of Mormon.
Henry A. Matis, Conference Report, April 1955, p.10 wrote:
I bear testimony to you, brethren, that I know this Church is true. I know that Joseph Smith was and is a prophet of our Heaven sent to restore the Gospel truths on this earth.
Bruce R. McConkie, Conference Report, April 1961, p.39 wrote:
Before baptism a person must believe that this Church is true; that it is in fact the Lord's earthly kingdom; that the priesthood and keys are here; that those who now officer it are legal administrators sent of God to preach the gospel and administer in the ordinances thereof.
Henry D. Moyle, Conference Report, April 1961, p.100 - p.101 wrote:
People by the thousands can be heard all over the earth to testify that the message of the missionaries of the Church is true.
Franklin D. Richards, Conference Report, October 1962, p.34 wrote:
The discussion plan is not designed to convince intellectually, but rather, the discussions are instruments through which the Holy Ghost can work to awaken a spiritual awareness and bring a personal testimony into the hearts of the listeners that Joseph Smith is a Prophet and the Church is true.
A priest from Ben Lomond Stake, Conference Report, October 1963, p. 75 wrote:My testimony is strong. I know the story of this great Church is true.
Mark Peterson,Conference Report, October 1968, p.94 wrote:
I would like to conclude tonight by saying that I know this Church is true, that the priesthood we hold is truly the priesthood of our Heavenly Father, and that if we continue in this partnership in the manner in which we should, our Heavenly Father will help us and bless us in everything we endeavor to do.
Bishop Victor L. Brown, Conference Report, October 1968 wrote: In his own words, this change is all because of his knowledge now that God lives, that the Church is true, and that there is a living prophet on the earth today. He didn't say so, but it is my feeling that this change is noticeable because of the peace of mind this young man now has that he didn't have before.
Bishop Victor L. Brown, Conference Report, October 1968, p.121 wrote:
I heard David O. McKay say that the highest ideal ever taught in any religion is service to our fellowman. This I believe with all my heart, and I know I should be thankful that Scott was baptized before he was killed, and I truly am."
Then he continued: "Brethren, I know the Church is true. I know that this Church is divinely directed, that Joseph Smith had revelations, and that Jesus Christ is the head of the Church today. I pray that while we are over here we might strive to build our characters, to build our testimonies, that we might all do a little bit of suffering so that we might all know the joy that comes from just plain living, and that as long as our hearts are beating, and we're walking and talking, we don't have a thing to complain about, not at all."
Delbert L. Stapley, Conference Report, April 1969, p. 47 wrote:This church is true; it has value and is meaningful to those seeking exaltation and eternal life. If this church is worth anything, it is worth everything! There is no exaltation and eternal glory without it.