"For as I passed by, and beheld, I found an altar with this inscription, TO THE UNKNOWN GOD. Whom therefore ye ignorantly worship, Him I declare unto you." Acts 17:23
Even today there are many, many people that worship an unknown god. They have a sense that there is a God and they see through creation that there must be a God, but they have not yielded to the Holy Spirit to receive God. Paul preached the God of the universe to these people. It is your duty and mine to also tell others about having met God. After all, you and I also worshipped or acknowledged the unknown God at one time in our lives.
It's not that unusual if the unbelieving world doesn't recognize God. But when those in the Church act like they don't know God either... You can have religion and still not know God. Paul did the right thing. He simply declared God to the people. I think we try too many alternate methods of evangelizing people, when we should be telling them about our own personal daily experience with God.
People will believe a genuine report of an actual event. I tell people, "I was talking with God and asking about a certain issue and God spoke to me clearly, that I was on the right path and to pursue it a little further." Just be honest. There were times when I also had to say, "I have been trying to hear the voice of God for days, but it just seems like I just can't make contact."
The truth is the conduit for a genuine believable testimony. You might very well be the person that God chose, to cross paths with someone, that has been looking for a genuine answer to their own lifetime quest. Today might be the day.