There is no recorded story of James or any of Jesus' other brothers believing in Christ prior to his death. Scriptural evidence seems to support that idea that James became a follower of Christ following a first-hand encounter with the resurrected Jesus. Far from being an obscure witness, James went on to become an apostle, lead the early church in Jerusalem, and die as a Christian martyr (as recorded by first century Jewish historian Josephus). The unbelief of Jesus' family was likely awkward for the early church and makes the late conversion and eyewitness account of his brother James especially significant.
1 CORINTHIANS 15: 5-7
"...he appeared to Cephas, then to the Twelve. Then he appeared to over five hundred brothers and sisters at one time; most of them are still alive, but some have fallen asleep. Then he appeared to James, then to all the apostles."
There are three possible alternatives:
Now when she had said this, she turned around and saw Jesus standing there, and did not know that it was Jesus. Jesus said to her, 'Woman, why are you weeping? Whom are you seeking?' She, supposing him to be the gardener, said to him, 'Sir, if you have carried him away, tell me where you have laid him, and I will take him away.' Jesus said to her, 'Mary!' She turned and said to him, 'Rabboni!' (which is to say, Teacher).
Mary The Mother Of James, Salome, And Joanna
And behold, Jesus met them and greeted them. And they came up and took hold of his feet and worshiped him.
The Lord is risen indeed, and has appeared to Simon!
Two Disciples On The Emmaus Road
And behold, two of them were going that very day to a village named Emmaus, which was about seven miles from Jerusalem. And they were conversing with each other about all these things which had taken place. And it came about that while they were conversing and discussing, Jesus himself approached, and began traveling with them. But their eyes were prevented from recognizing him.
All Of The Disciples
And after eight days his disciples were again inside, and Thomas with them. Jesus came, the doors being shut, and stood in the midst, and said, 'Peace to you!' Then he said to Thomas, 'Reach your finger here, and look at my hands; and reach your hand here, and put it into my side. Do not be unbelieving but believing.' And Thomas answered and said to him, 'My Lord and My God!'
Seven Disciples At The Sea Of Galilee
After these things Jesus manifested himself again to the disciples at the Sea of Tiberias, and he manifested himself in this way. There were together Simon Peter, and Thomas called Didymus, and Nathanael of Cana in Galilee, and the sons of Zebedee, and two other of his disciples.
A Mountain In Galilee
But the eleven disciples proceeded to Galilee, to the mountain which Jesus had designated. And when they saw him, they worshiped him; but some were doubtful.