Can an Unbeliever Exercise Faith?

Definitely!

If an unbeliever could not exercise faith, he could never become a believer, and everyone would go to hell!

All the people in the four gospels who got healed by “their faith” were unbelievers, as no one could be born again yet at that time!

All the Old Testament examples in the Hebrews 11 Faith Hall of Fame were unbelievers.

Jesus said that WHOSOEVER speaks to a mountain or tree and commands it to move and believes in his heart that what he says will come to pass, he will have whatever he says.  This is not limited to believers, nor is it limited to good things!  Death and life are in the power of the tongue (Proverbs 18:21) for both believers and unbelievers – that verse wasn’t taken from a New Testament epistle to the church, so it also applied to unregenerate people under the Old Covenant.  The tongue can set the course of nature on fire even for a believer (James 3:6)!

It is quite typical for an unbeliever to believe and say that he is going to die young of cancer like some relative(s) did, and his misplaced faith (in the devil’s ability rather than God’s ability) will see him to an early grave.

God has given EVERY man, not just believers, the measure of faith (Romans 12:3).  Therefore, EVERYONE can believe.