Answer: It is doubtful that Paul and Timothy were good friends right after they met. Paul (called Saul before his conversion), by his own admission, persecuted Christians. He stated this fact rather bluntly before a large crowd at Jerusalem's temple who wanted to kill him!
"I (Paul) am a Jew . . . brought up here in Jerusalem as a student of Gamaliel. I persecuted to the death the people who followed this Way (the teachings of Jesus Christ). I arrested men and women and threw them into prison . . . (Acts 22:3 - 4)
Saul (Paul), when Stephen was martyred by being stoned to death, held the garments of those who murdered him (Acts 7:58). Whether Timothy was a Christian at this time we do not know, but if he was it is obvious there would not have been any friendship between them.
Paul and Timothy first met during his second missionary journey, which began in late Autumn 49 A.D. After visiting and preaching the gospel in Tarsus and Derbe with fellow evangelist Silas, Paul met Timothy in Lystra (likely in his home). Timothy's father was a Greek (a Gentile). Although born Jews, both his mother Eunice and grandmother Lois were converted believers (2Timothy 1:5).
Timothy traveled with Paul on most of his second journey, served him in Ephesus (Acts 19:22) and was with him during his imprisonment in Rome (Philemon 1:1). The two grew so close that the apostle considered him "a true son in the faith" (1Timothy 1:2).
Just a few months before his martyrdom, while a prisoner in Rome, Paul wrote his last epistle. This letter, known in the Bible as 2Timothy, was dedicated to his friend Timothy. In it he expressed his respect and love for his fellow evangelist and traveling companion.
I thank him (God) as I remember you always in my prayers night and day. I remember your tears, and I want to see you very much, so that I may be filled with joy. I remember the sincere faith you have, the kind of faith that your grandmother Lois and your mother Eunice also had. I am sure that you have it also (2Timothy 1:3 - 5).
Paul died by beheading around May of 68 A.D. Tradition states that Timothy also died a martyr in 97 A.D.