The spring or fountain on Mt Helikon, which resulted from the blow of Pegasos' hoof, and is called Hippokrene, "Horse-Fountain," seems to be the most famous one in Greek mythology. I'm unaware of any on Mt Olympos.
The fountain or well called Peirene, located on the Akrokorinthos (the citadel of the city of Korinthos [Corinth]), was believed to have arisen from the tears which Peirene, the Naiad (water-nymph) of the region, shed in grief at the death of Kenkhrias (her son by Poseidon), who was somehow unintentionally killed by his cousin Artemis.
On Mt Ithome in Messenia there was a fountain called Klepsydra, whose name supposedly has something to do with theft, since it is said that Rhea, from fear of her husband Kronos, organised for the Kouretes (Curetes) to "steal" her newborn son Zeus away. The Kouretes entrusted Zeus to the care of the Messenian Oceanides Ithome and Neda, who bathed him in the spring or fountain which was, from this incident, named Klepsydra.
There's also a story about the Sicilian fountain or spring of Kyane (Cyane), named for a water-nymph who was married to the local river Anapos and who was one of the playmates of the young goddess Korē. The most well-renowned nymph in all of Sicily at the time, Kyane dwelt in her pool or fountain of water in the meadows of Enna (or Henna) on Syracuse Island off the coast of Sicily's mainland, where Korē liked to chill out the most, since she owned all this as her portion from the divisions of the world amongst the deities of Mt Olympos. When Haides (Hades) was abducting Korē to take her down to the Underworld, Kyane heard her cries and rose waist-high out of her pool of water, tried to bar Haides from his course and begged him not to ravish the goddess away but to woo her instead of taking her against her will. But Haides angrily thrust his sceptre into the ground with such force that the violence of the blow caused the bed of Kyane's pool to break open and thus injure the pool, which was the nymph herself. Haides then opened there and then a road to his kingdom, the Underworld, and disappeared down it with his niece Korē. The opening then closed up again. In grief at the loss of her companion and in anguish at the violation of her pool, Kyane dissolved into yet more water after pining away on the same spot. Korē’s girdle had been left floating on the surface of the pool and became the only physical evidence, apart from the violence done to the nymph’s pool, that that was the exact location of Korē’s abduction. Centuries later, while carrying out his tenth task under the servitude of Eurystheus, which task was to steal the oxen of Geryones, Herakles (Hercules) was making a circuit of Sicily when, as he was passing through Syracuse Island, he happened by this pool and heard the story of Korē’s abduction there in Enna, whence he offered sacrifices on a magnificent scale to his aunt, Korē’s mother Demeter, and to Persephone, the Underworld goddess that Kore had become, who was his half-sister, cousin and now also aunt. To Persephone he offered the largest bull of the entire herd of Geryones’ oxen by casting it into the pool of Kyane. Then, by commanding the natives to there conduct an annual gathering and the same sacrifice in splendid fashion, he instituted this festival among the people of Syracuse on the island at the spot of the event, at which a bull was sunk into the well as a sacrifice. Private individuals, however, when the ceremony was not on behalf of the entire community, offered lesser victims. The well, or fountain, became sacred to Persephone.