While the terminology is of later advent, consideration of the following scripture reveals that this is actually a biblical concept:

Or do you not know that your body is a temple of the Holy Spirit within you, whom you have from God? You are not your own, for you were bought with a price. So glorify God in your body. - 1 Corinthians 6:19-20 ESV

We were designed for the Spirit's habitation, but like Samson (cf. Judges 16:20), we all too easily overlook the emptiness that denotes "The Glory has departed" (cf. 1 Samuel 4:21). For those who have been filled with the Spirit, there can be no mistake that their prior experience must necessarily be characterized as 'emptiness' in comparison to:

Whoever believes in me, as the Scripture has said, ‘Out of his heart will flow rivers of living water.’ - John 7:38 ESV

and hope does not put us to shame, because God's love has been poured into our hearts through the Holy Spirit who has been given to us. - Romans 5:5 ESV

Elsewhere (cf. Matthew 12:43-45), Jesus warns on the danger of the empty state.

Pascal of course, knew what he was talking about.

There is a variant paraphrased quotation of Pascal floating all over the internet which reads:

"There is a God shaped vacuum in the heart of every man which cannot be filled by any created thing, but only by God, the Creator, made known through Jesus."

Some have pointed this out and taken issue with it: Incorrect Pascal Quotes.

Now I am not sure where this variant paraphrase (of the actual quote offered above) originated nor how long ago. It seems well established and well formed. However it also does give the idea a definite Christian spin. If it is old enough it could certainly be the source of the idea under consideration.

It may have actually come from a C.S. Lewis quote in Mere Christianity:

If I find in myself a desire which no experience in this world can satisfy, the most probable explanation is that I was made for another world.

This speaks of the inner desire within us that is unsatisfied by everything in this world.

Romans 1:20-23 is the Scripture passage I always think of when I hear that phrase.

For since the creation of the world God’s invisible qualities—His eternal power and divine nature—have been clearly seen, being understood from what has been made, so that people are without excuse. For although they knew God, they neither glorified Him as God nor gave thanks to Him, but their thinking became futile and their foolish hearts were darkened. Although they claimed to be wise, they became fools and exchanged the glory of the immortal God for images made to look like a mortal human being and birds and animals and reptiles.

Also Ecclesiastes 3:11b

He has also set eternity in the human heart; yet no one can fathom what God has done from beginning to end.

Thomas, Pascal and CS Lewis are paraphrasing Augustine.

Fecisti nos ad Te, et inquietum est cor nostrum, donec requiescat in Te.
— St Augustine, Confessions 1.1.

Translated by Maria Boulding as,

You stir us so that praising you may bring us joy, because you have made us and drawn us to yourself, and our heart is unquiet until it rests in you.

O’Donnell comments:

inquietum . . . requiescat: This initial disquiet is answered by the adumbration of eternal rest at 13.35.50 - 13.38.53, the last lines of the text; cf. also 1.5.5, ‘quis mihi dabit adquiescere in te?’ This restlessness arises from disorder: 13.9.10, ‘minus ordinata inquieta sunt; ordinantur et quiescunt.’ For ordo, see on 1.7.12 and see further on 13.9.10. Cf. en. Ps. 38.5, ‘coepit esse inquietum cor meum. . . . et suspirans in finem quendam, ubi ista non erat passurus, in illum, inquam, finem quo dicturus est bono erogatori dominus, “intra in gaudium domini tui.”’ (Mt. 25.21: cf. 9.10.25, where the same scriptural quotation is the culmination of the Ostia vision); cf. also en. Ps. 91.2, 48. s. 2.6. This phrase has evoked an abundant literature: A. Di Giovanni, L’inquietudine dell’anima (Rome, 1964), esp. 87n8; A. Pincherle, Augustinus 13(1968), 353-368 (on requies and the link to the last pages of conf.); E. Maccagnolo, Riv. di Filos. Neo-scolastica 71(1979), 314-325; G. Lawless, REAug 26(1980), 45-61 (on ‘interior peace’); and generally de la Peza (see next note).

See also

We all want to live happily; in the whole human race there is no one who does not assent to this proposition, even before it is fully articulated.
— St Augustine, De moribus eccl. 1,3,4: PL 32,1312.

How is it, then, that I seek you, Lord? Since in seeking you, my God, I seek a happy life, let me seek you so that my soul may live, for my body draws life from my soul and my soul draws life from you.
— St Augustine, Confessions 10,20: PL 32,791.

The concept is biblical:

John 7:37 Now on the last day, the great day of the feast, Jesus stood and cried out, saying, "If any man is thirsty, let him come to Me and drink. 38 "He who believes in Me, as the Scripture said, 'From his innermost being shall flow rivers of living water.'" 39 But this He spoke of the Spirit, whom those who believed in Him were to receive; for the Spirit was not yet given, because Jesus was not yet glorified. NAS

The word "innermost being" is the Greek koilos, meaning hole or empty place. It is often translated belly or womb. In Jn 7, Jesus is describing a spiritual empty place in the heart of man, not a physical belly or womb. This empty place is the source of thirst and the divine solution to this hunger/thirst is to fill it with living water from the Holy Spirit.

Perhaps the notion goes back to St. Augustine of Hippo who wrote in his Confessions:

You have made us for yourself, O Lord, and our hearts are restless until they rest in you.

St. Augustine was a very rebellious man who took a long time to come around to Christianity and became a Doctor of the Church.

This is a great question, as in the fact that trying to trace these bibical origins are hard. First, let us look at the Acts 17: 22-27:

22 Then Paul stood before the meeting of the Areopagus and said, "People of Athens, I can see you are very religious in all things. 23 As I was going through your city, I saw the objects you worship. I found an altar that had these words written on it: TO A GOD WHO IS NOT KNOWN. You worship a god that you don't know, and this is the God I am telling you about! 24 The God who made the whole world and everything in it is the Lord of the land and the sky. He does not live in temples built by human hands. 25 This God is the One who gives life, breath, and everything else to people. He does not need any help from them; he has everything he needs. 26 God began by making one person, and from him came all the different people who live everywhere in the world. God decided exactly when and where they must live. 27 God wanted them to look for him and perhaps search all around for him and find him, though he is not far from any of us.

As you can see, when God created humans, he wanted them to look around for him and search for him. This is one of God's plans, and in doing so, he created this "hole" that cannot be filled by any other: in short, that "God-shaped hole" was created by God for us to look for him.

I have another verse as example; let us look at Ecclesiastes 3:10-12:

10 I have seen the burden God has laid on the human race. 11 He has made everything beautiful in its time. He has also set eternity in the human heart; yet no one can fathom what God has done from beginning to end. 12 I know that there is nothing better for people than to be happy and to do good while they live.