I have a cousin who's like this. I think he's a person who denies that God exists.
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Lack of humility. Too much Pride. 2 Chronicles 7:13-15: "When I shut up heaven and there is no rain, or command the locusts to devour the land, or send pestilence among My people, if My people who are called by My name will humble themselves, and pray and seek My face, and turn from their wicked ways, then I will hear from heaven, and will forgive their sin and heal their land. Now My eyes will be open and My ears attentive to prayer [made] in this place." Any person who wants to obtain heaven has to humble themselves at the foot of the cross, proclaiming that Jesus died in their place and they can not achieve entrance by any work of their own effort. In this day, humility, real humbleness is missing, even in the church. Notice God says, "My people called by My Name.” That would be everyone who calls upon the Name of Jesus. God helps us to be humble before Him. Philippians 2:9-11: "Therefore God also has highly exalted Him and given Him the name which is above every name, that at the name of Jesus every knee should bow, of those in heaven, and of those on earth, and of those under the earth, and [that] every tongue should confess that Jesus Christ [is] Lord, to the glory of God the Father." Humility willingly now or forced in the future. It's really sad. I have known many who have died without submission to Jesus. For those of us who know the reality of hell, we should truly grieve for the lost. Jesus knew the reality of hell and spoke about it more often that any other in the Bible. Seven times in Revelation 2 & 3, Jesus says "whoever overcomes". Overcoming pride is horribly difficult, but Jesus gives us the power by His Holy Spirit to do it.
I can't address the specific case of the questioner's cousin, but I would say in general that the reason many people refuse to believe in God (no matter how much evidence one might present to them) is because of pride, which is the sin that caused Lucifer the angel to rebel against God, to become Satan the devil, and to be cast out of heaven (Isaiah 14:12-15; Luke 10:18). People by nature do not want to submit themselves to a higher Being, or even to acknowledge that Being's existence, because they view such submission as an unacceptable infringement on their own autonomy and control of their lives (saying, in effect, "Nobody's going to tell ME what to do!"). The Christian apologist C. S. Lewis referred to pride as "The Great Sin", and had an excellent discussion of it in his book Mere Christianity, which can be viewed in full for free at https://www.dacc.edu/assets/pdfs/PCM/merechristianitylewis.pdf
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