The story of King Talut, Prophet David and Goliath is mentioned in the Glorious Quran as follows:
002.246 Hast thou not Turned thy vision to the Chiefs of the Children of Israel after (the time of) Moses? they said to a prophet (That was) among them: "Appoint for us a king, that we May fight in the cause of God."
He said: "Is it not possible, if ye were commanded to fight, that that ye will not fight?"
They said: "How could we refuse to fight in the cause of God, seeing that we were turned out of our homes and our families?" but when they were commanded to fight, they turned back, except a small band among them. But God Has full knowledge of those who do wrong.
002.247 Their Prophet said to them:
"God hath appointed Talut as king over you."
They said: "How can he exercise authority over us when we are better fitted than he to exercise authority, and he is not even gifted, with wealth in abundance?"
He said:
"God hath Chosen him above you, and hath gifted him abundantly with knowledge and bodily prowess: God Granteth His authority to whom He pleaseth. God careth for all, and He knoweth all things."
002.248 And (further) their Prophet said to them:
"A Sign of his authority is that there shall come to you the Ark of the covenant, with (an assurance) therein of security from your Lord, and the relics left by the family of Moses and the family of Aaron, carried by angels. In this is a symbol for you if ye indeed have faith."
002.249 When Talut set forth with the armies, he said:
"God will test you at the stream: if any drinks of its water, He goes not with my army: Only those who taste not of it go with me: A mere sip out of the hand is excused." but they all drank of it, except a few. When they crossed the river,- He and the faithful ones with him,- they said: "This day We cannot cope with Goliath and his forces." but those who were convinced that they must meet God, said: "How oft, by God's will, Hath a small force vanquished a big one? God is with those who steadfastly persevere."
002.250 When they advanced to meet Goliath and his forces, they prayed: "Our Lord! Pour out constancy on us and make our steps firm: Help us against those that reject faith."
002.251 By God's will they routed them; and David slew Goliath; and God gave him power and wisdom and taught him whatever (else) He willed. And did not God Check one set of people by means of another, the earth would indeed be full of mischief: But God is full of bounty to all the worlds.
002.252 These are the Signs of God: we rehearse them to thee in truth: verily Thou art one of the apostles.
(This text is taken from the English translation by Abdullah Yusuf Ali of the original Quran in Arabic.)