Yes, it's absolutely possible for Spock, and any other alien hybrid, to have children. Several relevant episodes of Enterprise are referenced in the comments, which I want to expand on.
The episode Future Tense revolves around the Enterprise crew discovering a ship from the future. The pilot's corpse is found inside, and Doctor Phlox determines that it was the result of many generations of inter-species breeding. In particular, it had DNA from Humans, Vulcans, and Terrellians, as well as others that Phlox couldn't identify.
A later Enterprise episode, E2, involves a future version of Enterprise run by the descendants of the original crew. The crew of this ship is about two generations removed from the main cast and the First Officer, Karyn Archer, is Captain Archer's grand-daughter. Karyn's grandmother is specifically said to be an alien, which means that Karyn's father was a half-Human who successfully reproduced.
In a more concrete example, Worf has a son by K'Ehleyr, introduced in the TNG episode The Emissary. K'Ehleyr constantly points out that she's only half-Klingon, and yet she and Worf have a child with no problems.
A similar case occurs in Voyager: human Tom Paris and half-Klingon B'Elanna Torres have a daughter, born in the episode Endgame.
EDIT: As HorusKol points out in comments, the Next Generation episode The Drumhead involves Crewman Simon Tarses, who had a Romulan grandfather.
The Star Trek mythology even explains how this is possible. A late episode of The Next Generation, The Chase establishes that all humanoid species in the galaxy share common genetic material with a progenitor race, who specifically seeded the galaxy to create different humanoid species:
Although it kind of flies in the face of known genetics, it's a good enough explanation for Star Trek.