The Bible speaks of fruit in several different ways. In the end, all spiritual fruit is the result of God's Spirit working in the life of a believer. As you mentioned, there is the 'fruit of the Spirit' described in Galatians 5:22-23. These are the attributes of Jesus manifested in the life of a believer. Romans 6:22 speaks of the 'fruit of holiness.' Philippians 1:11 speaks of the 'fruits of righteousness.' Colossians 1:6 speaks of the fruit brought forth by the gospel, which seems to be people coming to God through the new birth.
Jesus said, 'Every tree that does not bear good fruit is cut down and thrown into the fire' (Matthew 7:19). The bottom line is this: When Jesus fills us with His Spirit, He intends that we would submit to that Spirit and allow the Spirit to bring forth its fruit in our lives. We cannot produce the fruit by our own efforts; we can only submit to the Spirit. If there is no fruit, that fact alone is the evidence that indicts us for failing to make Jesus our Lord. Just as apple trees were created to produce apples, Christians are created to produce the spiritual fruit of Christ.