What is a quote from the ghost of old Hamlet or young Hamlet that illustrates that Hamlet is motivated by religion?

Hamlet is motivated by religion because Hamlet is scared of losing grace in the eyes of God; the ghost tells him that he has to kill Claudius in order to stop the sin of incest from taking place, however, which could make the murder justified before God.
It's clear early on that Hamlet himself is motivated by religion; it guides many of his choices. In act 1, scene 2, Hamlet says,

Oh, that this too, too sullied flesh would melt,Thaw, and resolve itself into a dew,Or that the Everlasting had not fixedHis canon 'gainst self-slaughter!

He's expressing his desire to commit suicide because of the death of his father and subsequent marriage of his mother. Yet it's clear from his words that he won't kill himself, because God has commanded men not to do so. He is a religious man who believes in the word of God.
When the ghost says, "My hour is almost come / When I to sulfurous and tormenting flames / Must render up myself," it's another confirmation for Hamlet that he shouldn't go against God's wishes. He said earlier in the play that his father was perfect in every way—and yet the ghost who Hamlet thinks might be his father is spending his afterlife in torment and believes he was a sinful man. It's a sign to Hamlet that he shouldn't go against God's commandments for simple revenge.
It's the sin of incest—along with vengeance—that guides the ghost's requests to Hamlet. Hamlet himself is also tormented by the incest taking place between his mother and uncle. Part of his anguish is because of the unnatural relations; it might be amplifying the grief he feels over the death of his father.
The ghost implores Hamlet to let God take care of his mother—even while he's urging him to take revenge on his uncle and stop what he sees as unnatural acts. He says,

If thou hast nature in thee, bear it not.Let not the royal bed of Denmark beA couch for luxury and damnèd incest.But howsoever thou pursuest this act,Taint not thy mind, nor let thy soul contriveAgainst thy mother aught. Leave her to heavenAnd to those thorns that in her bosom lodgeTo prick and sting her.

This is more proof that both Hamlet and the ghost are guided by religion. He chooses not to injure his mother during his quest for revenge, even though he is furious at her actions. The ghost believes that if Hamlet kills Claudius, he will be justified in doing so to stop the sin of incest.
At the end of the play, when Gertrude is dead and Hamlet confronts Claudius, he brings up the sin of incest again. This is more evidence that he was guided by his religious beliefs to tenaciously pursue Claudius's death. Hamlet says,

Here, thou incestuous, murderous, damnèd Dane,Drink off this potion. Is thy union here?Follow my mother.

Claudius drinks and dies. His mission fulfilled, Hamlet assures the dying Laertes that he, Hamlet, will follow Laertes to heaven. This means that he trusts in the will of God and the word of the ghost and that he was guided by the correct motivations for killing Claudius. A religious man would be unlikely to be so confident in his ascension to heaven after just killing someone unless he was sure it was the right, godly thing to do.
Hamlet never seems to doubt the presence and wisdom of God. Even after everything he's learned and now understands, in act 5, scene 2, Hamlet says,

And praised be rashness for it: let us knowOur indiscretion sometimes serves us wellWhen our deep plots do pall, and that should teach usThere’s a divinity that shapes our ends,Rough-hew them how we will—

This means that God is watching over us even when we mess up. Hamlet doesn't feel that he's done everything right; if he had, Ophelia wouldn't be dead. Claudius wouldn't still be alive. Still, though, he's motivated by God to continue despite the tragedy because he truly believes he's doing the right thing.
Once almost everyone is dead at the end of the play, Horatio comments on Hamlet, saying,

Now cracks a noble heart. Good night, sweet prince, And flights of angels sing thee to thy rest!

If Horatio is right, then Hamlet—whatever he may have done—is redeemed in the eyes of God. He was guided by his religious beliefs and the word of the ghost. In the end, he may have murdered a man, but he doesn't believe that he is damned for doing so.