There's at least one very famous example of a Greek being charged, ostensibly because he didn't conform to religious expectations: Socrates. Plato's Apology describes Socrates' defense in the Athenian court against the charges of impiety and corrupting the youth. Socrates neatly summarizes the charge against him in 24b (this is a notation system for pointing to specific passages in Plato):

It [that is the charge against Socrates] goes something like this: Socrates is guilty of corrupting the young and of not believing in the gods in whom the city believes, but in other new spiritual things. Such is their charge.

The Apology describes that Socrates was ultimately declared guilty and given the penalty of death (35d-36b). Another dialogue of Plato, the Phaedo, describes Socrates' drinking the hemlock that killed him.

Plato himself was skeptical of some traditions about the gods. In book II of the Republic, Plato describes how he would have the guardians of his idealized city be educated—and in the process he points to a number of poetic texts about the gods that he would not permit to be taught in the city. One example of this appears in 381a-d, where Plato (using the lips of Socrates in the dialogue) articulates the notion that a god cannot be changed or altered at all, then says...

Then let no poet tell us about Proteus or Thetis, or say that, "The gods, in the likeness of strangers from foreign lands, adopt every sort of shape and visit our cities."

That's a quotation from the Odyssey that Plato is saying ought not to be taught in his idealized city. But we have no evidence that Plato faced any trouble for questioning such a venerable text as the Odyssey.