Answer: It is doubtful that Paul and Timothy were good friends shortly after their first meeting. The apostle, before his conversion, had a fanatical zeal that didn't exactly lend itself to people easily bonding with him. He admitted to his over-the-top personality to a group of Jews at Jerusalem's temple.
"I 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), even after becoming a Christian, still retained a personality that was polarizing and intense, which made being his friend a bit difficult. It would take time for the apostle to mellow out a bit in order for a closer relationship between him and Timothy to form.
Paul first met his best friend 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, He met Timothy in Lystra where he likely discovered his father was a Greek (a Gentile) but his mother was a Jewess (2Tim. 1:5). Both her (Eunice) and her mother (Lois) had converted to Christianity.
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). Over time, the two grew so close that the apostle considered him "a true son in the faith" (1Tim. 1:2).
Paul, just a few months before his martyrdom while he was a prisoner in Rome, wrote his last epistle. This personal letter was dedicated to his friend Timothy. In this epistle 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 (2Tim. 1:3 - 5).
Paul died through beheading around May of 68 A.D. Tradition states his good friend Timothy died a martyr, 29 years later, in 97 A.D.