The phrase doesn't refer to the television/movie sci-fi franchise, the brand of bicycles nor the Boers' "Great Trek" inland migration in southern Africa in the 1830s and '40s.
In the LDS Church vernacular, "trek" means a modern-day re-enactment of the Mormon pioneers who crossed the Great Plains and Wyoming's Continental Divide en route the Salt Lake Valley.
Almost always those youth-oriented re-enactments use the pulling of smaller handcarts across rugged stretches of land rather than full-size wagons drawn by horses, mules or oxen. It's for the same reason LDS Church leaders tried the handcart mode with 10 pioneer companies in the 1850s — because of the cost and logistics of arranging for the livestock and the larger wagons.
As part of trek re-enactments over the past decade and a half, LDS youths and adults wearing period attire have struggled to pull the handcarts over rocks and ridges and through sage and sand, all the time learning of the sacrifices and struggles of 19th century pioneers — more specifically the ill-fated 1856 Martin and Willie handcart companies, revered for their faithfulness and sacrifice.