Calling upon the name of God is mentioned several times in the Book of Mormon. One reference particularly stands out to me:
27 Thus we may see that the Lord is merciful unto all who will, in the sincerity of their hearts, call upon his holy name. (Helaman 3)
"Thus we may see" is typically Mormon's way of pointing out important lessons we should learn from the verses immediately preceding this kind of statement. So let's look at the verses leading up to this statement, along with a few verses which come after and offer additional insights about what lessons we can learn:
24 And it came to pass that in this same year there was exceedingly great prosperity in the church, insomuch that there were thousands who did join themselves unto the church and were baptized unto repentance.
25 And so great was the prosperity of the church, and so many the blessings which were poured out upon the people, that even the high priests and the teachers were themselves astonished beyond measure.
26 And it came to pass that the work of the Lord did prosper unto the baptizing and uniting to the church of God, many souls, yea, even tens of thousands.
27 Thus we may see that the Lord is merciful unto all who will, in the sincerity of their hearts, call upon his holy name.
28 Yea, thus we see that the gate of heaven is open unto all, even to those who will believe on the name of Jesus Christ, who is the Son of God.
29 Yea, we see that whosoever will may lay hold upon the word of God, which is quick and powerful, which shall divide asunder all the cunning and the snares and the wiles of the devil, and lead the man of Christ in a strait and narrow course across that everlasting gulf of misery which is prepared to engulf the wicked—
30 And land their souls, yea, their immortal souls, at the right hand of God in the kingdom of heaven, to sit down with Abraham, and Isaac, and with Jacob, and with all our holy fathers, to go no more out. (Helaman 3)
It seems this passage, in addition to revealing the underlying meaning of "call[ing] upon [the name of God]," also connects other phrases with making and keeping sacred covenants. I've highlighted these in various colors above. Each of these can be connected to covenant symbols found in other parts of the Book of Mormon (especially on the small plates).
"gate of heaven" -- 2 Nephi 9:41; 2 Nephi 31:9, 17-18; 2 Nephi 33:9
"lay hold upon the word of God" -- 1 Nephi 8:24, 30; 1 Nephi 11:24-25; 1 Nephi 15:24-25
"strait and narrow course" -- 1 Nephi 8:20
"right hand of God" -- Mosiah 5:9 ("for he shall know the name by which he is called; for he shall be called by the name of Christ"); Mosiah 26:23-24 ("in my name they are called"); Alma 5:58 ("the word of God must be fulfilled"); Ether 12:4; Moroni 9:26
"kingdom of heaven" -- Alma 11:37; Helaman 5:32; 3 Nephi 14:21
"sit down with Abraham, Isaac, and Jacob...to go no more out" -- Alma 7:25; 3 Nephi 4:30; Mormon 9:11-13; 1 Nephi 17:40
Variants:
"enter in at the strait gate, and continue in the way which is narrow" Jacob 6:11
"lay hold upon every good thing" -- Moroni 7:19-21, 25 (see this post for more detail about how "every good thing" relates to covenants)
"lay hold upon the gospel of Jesus Christ" -- Mormon 7:8
Conclusion
This is an excellent example of what I believe to be a pattern throughout the Book of Mormon -- covenants are spoken of through symbolic language rather than at the surface level. It isn't until you attempt to connect one passage with others that you see numerous clear and consistent references to the importance of covenants.