That depends on what we mean when we say prove. If we mean “is it possible to present solid, compelling, and logical reasons to believe in the existence of God,” then the answer is yes. But if we mean “can God’s existence be demonstrated beyond all possible doubt,” then the answer is no.
A “no” answer should not cause those who believe in God to panic. Those who deny God’s existence cannot prove their position, either.
Some things are just beyond our ability to prove, and yet we accept them as true. I cannot prove that my wife loves me, but I’m pretty sure she does. I can’t prove that a breathtaking sunset is beautiful, but I know that it is. I can’t prove that torturing and murdering another human being is evil, but it is.
All of us deeply believe in things that can neither be proven or disproven, including the existence of God. And yet we find ourselves as certain about them as we are about the wind that blows in our faces.