There is nothing in the Word of God that requires believers to prove the existence of God.
In fact, the Bible itself never sets out to prove God's existence. It is always taken for granted from the very first page, which starts out saying, "In the beginning GOD..." No explanation, no introduction, no details...just GOD.
Furthermore, in the Psalms, the Holy Spirit moved upon David to write about God's existence saying, "The heavens declare the glory of God, and the sky above proclaims his handiwork." (Psalm 19:1) That verse tells us that we don't have to prove anything about the existence of God because the Creation itself is constantly proclaiming that He is. This "proclamation" is so convincing that the Apostle Paul wrote that all men are without excuse.
For what can be known about God is plain to them, because God has shown it to them. 20 For his invisible attributes, namely, his eternal power and divine nature, have been clearly perceived, ever since the creation of the world, in the things that have been made. So they are without excuse. (Romans 1:19-20)
The Bible speaks of defending and confirming the Gospel, (Philippians 1:7) but not God's existence. If someone chooses not to believe in a Creator God, it is because they have chosen to close their eyes to the evidence all around them.