There is no single verse declaring, "God is simple." But those who believe the doctrine believe that it is the only way to coherently hold onto all of the Bible's declarations about God. Catholicism and all the major Protestant confessions declare divine simplicity.
The blog post "Is divine simplicity scriptural?" says:
Simplicity is implicit in Scripture in that it follows from a strong doctrine of aseity and God's providence, which is found stated in John 1:3 and Colossians 1:16. These passages say that all things were created by God. So it is true that whatever is non-identical to God is created by God. But if God had parts he would have to create his parts, and in so doing create himself, which can’t be true since God is uncreated and uncaused. Hence, God doesn’t have any parts and is simple.
James Dolezal, author of God without Parts, similarly says:
[God] gives to all, but receives from none (Acts 17:25-25; Rom. 11:35-36).
Herman Bavinck writes in Reformed Dogmatics, Volume 2 (page 176):
If God is composed of parts, like a body, or composed of genus (class) and differentiae (attributes of different species belonging to the same genus), substance and accidents, matter and form, potentiality and actuality, essence and existence, then his perfection, oneness, independence, and immutability cannot be maintained.
Kevin DeYoung paraphrases Bavinck's point this way:
In other words, the simplicity of God not only prevents us from ranking certain attributes higher than others, it allows God to have "a distinct and infinite life of his own within himself" (Bavinck 177). He is not an abstract Absolute Idea who happens to have love, wisdom, and holiness, as if we first conceive of a being called God and then relate qualities to him. Rather, God in his very essence—within himself and by himself—is love, wisdom, and holiness. God is whatever he has. He is not the composite of his attributes, some in greater and some in lesser amounts. God is a simple being without parts or pieces. His attributes do not stick to him; he is what they are.
Therefore, the doctrine is built on the doctrines of God's aseity, providence, perfection, oneness, independence, immutability, self-sufficiency, and identity as creator. Each of these doctrines has a strong Biblical basis. From that list, oneness and immutability are the attributes most closely related to simplicity. Here are some of the verses that most strongly support the immutability and oneness of God:
Malachi 3:6 I the LORD do not change.
James 1:17 Every good gift and every perfect gift is from above, coming down from the Father of lights with whom there is no variation or shadow due to change.
Numbers 23:19 God is not man, that he should lie, or a son of man, that he should change his mind.
1 Samuel 15:29 The Glory of Israel will not lie or have regret, for he is not a man, that he should have regret.
Exodus 3:14 God said to Moses, "I am who I am." And he said, "Say this to the people of Israel, 'I am has sent me to you.'"
Deuteronomy 6:4 Hear, O Israel: The LORD our God, the LORD is one.
All scriptural citations are from the ESV.