I am interested in a source that is besides the book of Acts written by Luke. I am interested in reading about Paul's encounter on the road to Damascus in his own words.
Short version: Paul tells us the same basic story: he was persecuting Christians in Jerusalem, he was sent to Damascus, he was converted, he left Damascus, and he went to Arabia before going to see the church fathers and beginning his apostlehood. He also says he "has seen Jesus", which he doesn't link to a particular time but it makes sense in his overall narrative that seeing Jesus would be the moment of his conversion. He doesn't explicitly say he saw a blinding light, but some people find hints of his having been blinded and not fully recovered.
So it seems reasonable to assume Paul and Luke were on the same page, although Paul doesn't give us as many details as we might wish for, and not all in one place.
Here are some clues. All verses quoted in NIV.
In 1 Corinthians 9:1 he says he has seen Jesus, using a rhetorical question that suggests he's told them the story before:
Have I not seen Jesus our Lord?
As Ozzie notes, he says later in the same letter (1 Corinthians 15:8) that Christ "appeared to him".
In Galatians 1:13-17 he describes the persecution he committed, his going to Damascus, and his change of heart:
I persecuted the church of God and tried to destroy it ... When God ... was pleased to reveal his Son in me so that I might preach him among the Gentiles ... I went into Arabia. Later I returned to Damascus.
(Note: He doesn't say he was at Damascus in the first place, but it was clearly at the time he switched from persecutor to believer. Also, he goes on to recount more events described in Acts.)
In 2 Corinthians 11:32-33 he mentions his escape from Damascus in the same terms Luke does:
In Damascus the governor ... had the city of the Damascenes guarded in order to arrest me. But I was lowered in a basket from a window in the wall and slipped through his hands.
See what large letters I use as I write to you with my own hand!
Some people connect Paul's poor vision to the blinding light Luke describes and the scales that formed on his eyes, arguing that even after being healed, he never fully recovered his eyesight.
Thus, to my knowledge, Paul doesn't explicitly describe the event of Jesus' appearance as Luke does, but it would fit at the right time in his redemption narrative as he does tell it.
Does Paul mention anywhere in his own writings his encounter with Jesus and his coming to belief in the gospel?
Paul tells us that he was relieved of his heavy bloodguilt by God’s loving-kindness through Jesus Christ, who appeared to him even after his ascension to heaven: “
1 Corinthians 15:8-11 (NASB)
8 And last of all, as to one untimely born, He appeared to me also. 9 For I am the least of the apostles, [b]and not fit to be called an apostle, because I persecuted the church of God. 10 But by the grace of God I am what I am, and His grace toward me did not prove vain; but I labored even more than all of them, yet not I, but the grace of God with me. 11 Whether then it was I or they, so we preach and so you believed.
It’s my understanding that Galatians 1:12 is the key verse about Paul's encounter on the road to Damascus in his own words: ‘For I neither received it from man, nor was I taught it, but it came through the revelation of Jesus Christ.’ Also, Paul wrote in brief about his encounter in Ephesians 3: 2-3 saying, ‘... if indeed you have heard of the dispensation of the grace of God which was given to me for you, how that by revelation He made known to me the mystery (as I have briefly written already, by which, when you read, you may understand my knowledge in the mystery of Christ), ...’