Zeus is the Greek sky god, ruler of Olympus, and supreme arbiter among the gods, so he is most certainly immortal, and there are definitely stories that demonstrate this quality. To begin with, you might consider looking into the story of the birth of Athena, by which she emerges out of his head fully formed.

As an additional example, I would refer to a particular variation on the story of Zeus's battle with Typhoeus, sometimes referred to as Typhon, as related by Apollodorus (as cited in Barry B. Powell, Classical Myth, 4th Ed., Upper Saddle River, New Jersey: Pearson Education, 2004, pp. 92–93). This particular version of the story is interesting because in it, Typhon is actually able to defeat Zeus, and even cripple him, after which he hides Zeus in a cave, under guard by a dragon. But it's notable, as far as this answer is concerned, that, while he has defeated Zeus, he can't actually kill Zeus, and that this injury, while severe, is not itself permanently impairing. Hermes proceeds to sneak into the cave and assist his father, after which Zeus emerges healed. The fight with Typhon resumes, with Zeus victorious this time.

The one caveat you should be aware of, however, is that Greek mythology tends to be variable. Different places had their own traditions and stories, so we should expect a certain element of variance to emerge within the mythology. Nevertheless, I'd hope that, as far as this answer is concerned, these examples give some idea as to the vast degree of separation the Greeks understood as existing between mortals and the gods.

Zeus is immortal. This simply means that he cannot die, as opposed to mortals, who can. He is the god of sky and thunder, as well as lightning, law, order, and justice. He's the father of the gods in the Greek pantheon, the one whom all the other gods fear and respect. He's also the object of veneration from human beings, or mortals, who honor him with countless sacrifices and rituals. Zeus, like all the gods, regularly intervenes in human affairs—taking sides in conflicts, sending down bolts of thunder to destroy those mortals foolish enough to defy him, taking on the form of animals to allow him to mate with mortal females. He's certainly not someone to be trifled with, by either god or man. He is truly the first of the gods.