Infinity was a mathematical and philosophical concept of having no end.
In 2266, while considering the possibility of contamination from Psi 2000 causing unusual symptoms in several members of USS Enterprise's crew, Spock noted that space contained infinite unknowns, and therefore the tricorders might not have registered the source of the illness. (TOS: "The Naked Time")
In 2268, the brain of a disembodied Spock remarked that his body seemed to stretch into infinity. (TOS: "Spock's Brain")
When asked by Garth of Izar how he would respond to "queen to queen's level three", James T. Kirk told him there were potentially infinite possibilities. When Garth pushed him for an answer, Kirk pretended not to know which of them he wanted. (TOS: "Whom Gods Destroy")
In 2287, upon breaching the Great Barrier, Sybok explained to "God" that they sought his infinite wisdom. (Star Trek V: The Final Frontier)
In 2365, William T. Riker described the sight of a hole in space as being like looking into infinity. (TNG: "Where Silence Has Lease")
In 2367, following a dramatic increase in intelligence, Reginald Barclay and a hologram of Albert Einstein discussed infinity. (TNG: "The Nth Degree")
In 2373, Q proposed mating with Captain Kathryn Janeway to create a new breed of the Q species which would have the combined omnipotence and infinite intellect of Q with the best that Humanity had to offer. (VOY: "The Q and the Grey")