The Qur'an does not mention the act of stoning (Rajm) for any crime.

Stoning is called rajm (Arabic: رجم) in Islamic literature, and is a practice found in:

• Saudi Arabia

Stoning is in the Quran,

but it is only done by the disbelievers

Sleepers of Ephesus

[18:20] "If they discover you, they will stone you, or force you to revert to their religion, then you can never succeed.”  

Messengers

[36:18] They said, "We consider you bad omens. Unless you refrain, we will surely stone you, or afflict you with painful retribution.” 

Shu’aib

[11:91] They said, "O Shu`aib, we do not comprehend many of the things you are telling us, and we see that you are powerless among us. If it were not for your tribe, we would have stoned you. You have no value for us.”  

Noah

[26:116] They said, "Unless you refrain, O Noah, you will be stoned.”

Abraham

[19:46] He said, "Have you forsaken my gods, O Abraham? Unless you stop, I will stone you. Leave me alone.” 

[6:74] Recall that Abraham said to his father Ãzer, "How could you worship statues as gods? I see that you and your people have gone far astray.”

There is a Hadith that while the people were preparing the funeral for the prophet a goat came and ate the verses of the Quran having to do with stoning

[7:185] Have they not looked at the dominion of the heavens and the earth, and all the things GOD has created? Does it ever occur to them that the end of their life may be near? Which Hadith, beside this, do they believe in?

[6:19] Say, "Whose testimony is the greatest?" Say, "GOD's. He is the witness between me and you that this Quran has been inspired to me, to preach it to you and whomever it reaches. Indeed, you bear witness that there are other gods beside GOD." Say, "I do not testify as you do; there is only one god, and I disown your idolatry.”

[6:38] All the creatures on earth, and all the birds that fly with wings, are communities like you. We did not leave anything out of this book. To their Lord, all these creatures will be summoned.

Abrogation

https://www.youtube.com/watch?v=x41Oz3PpeRs

Q&A: Is Stoning Part of Islamic Sharia? | Dr. Shabir Ally

Abrogate = to abolish to indicate that certain verses no longer apply that they have been voided and replaced with something new

[2:106] When we abrogate any miracle, or cause it to be forgotten, we produce a better miracle, or at least an equal one. Do you not recognize the fact that GOD is Omnipotent?

2:106 word = ayat = miracle, sign and verse 

[17:101] We supported Moses with nine profound miracles—ask the Children of Israel. When he went to them, Pharaoh said to him, "I think that you, Moses, are bewitched."

[18:27] You shall recite what is revealed to you of your Lord's scripture. Nothing shall abrogate His words, and you shall not find any other source beside it.

Hadith & Sunna: Fabrications by the Prophet's Enemies

[6:112] We have permitted the enemies of every prophet—human and jinn devils—to inspire in each other fancy words, in order to deceive. Had your Lord willed, they would not have done it. You shall disregard them and their fabrications.

Important Criterion

[6:113] This is to let the minds of those who do not believe in the Hereafter listen to such fabrications, and accept them, and thus expose their real convictions.*

[6:114] Shall I seek other than GOD as a source of law, when He has revealed to you this book fully detailed?* Those who received the scripture recognize that it has been revealed from your Lord, truthfully. You shall not harbor any doubt. [6:115] The word of your Lord is complete,* in truth and justice. Nothing shall abrogate His words. He is the Hearer, the Omniscient. [6:116] If you obey the majority of people on earth, they will divert you from the path of GOD. They follow only conjecture; they only guess.

Abraham

[21:51] Before that, we granted Abraham his guidance and understanding, for we were fully aware of him. [21:52] He said to his father and his people, "What are these statues to which you are devoting yourselves?" [21:53] They said, "We found our parents worshiping them." [21:54] He said, "Indeed, you and your parents have gone totally astray." [21:55] They said, "Are you telling us the truth, or are you playing?" [21:56] He said, "Your only Lord is the Lord of the heavens and the earth, who created them. This is the testimony to which I bear witness. [21:57] "I swear by GOD, I have a plan to deal with your statues, as soon as you leave."

[21:58] He broke them into pieces, except for a big one, that they may refer to it. [21:59] They said, "Whoever did this to our gods is really a transgressor." [21:60] They said, "We heard a youth threaten them; he is called Abraham." [21:61] They said, "Bring him before the eyes of all the people, that they may bear witness." [21:62] They said, "Did you do this to our gods, O Abraham?"

Abraham Proves His Point

[21:63] He said, "It is that big one who did it. Go ask them, if they can speak." [21:64] They were taken aback, and said to themselves, "Indeed, you are the ones who have been transgressing." [21:65] Yet, they reverted to their old ideas: "You know full well that these cannot speak." [21:66] He said, "Do you then worship besides GOD what possesses no power to benefit you or harm you? [21:67] "You have incurred shame by worshiping idols beside GOD. Do you not understand?"

Profound Miracle

[21:68] They said, "Burn him and support your gods, if this is what you decide to do." [21:69] We said, "O fire, be cool and safe for Abraham.” [21:70] Thus, they schemed against him, but we made them the losers. [21:71] We saved him, and we saved Lot, to the land that we blessed for all the people. [21:72] And we granted him Isaac and Jacob as a gift, and we made them both righteous.