The disciples were first called Christians at Antioch is in the Bible. Acts 11:26.
That appellation wasn't derived by those faithful to Christ.
https://www.gotquestions.org/meaning-of-Christian.html
"In the New Testament, believers never refer to themselves as “Christians”; rather, they use such terms as
brethren (
Acts 15:1;
1 Corinthians 16:20, NAS),
disciples (
Acts 11:26;
14:24, NKJV), and
saints (
Acts 9:13;
2 Corinthians 13:13, ESV). Before his conversion,
Saul of Tarsus sought out those “who belonged to the Way” (
Acts 9:2), indicating that an early label for Christians could have been “people of the Way” (see also
Acts 19:9;
24:22)."
"
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What is the meaning of the term Christian?
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The followers of Jesus Christ were first referred to as “Christians” by the Gentiles of Syrian Antioch, and the name was more than likely meant as an insult (see Acts 11:26).
In the New Testament, believers never refer to themselves as “Christians”; rather, they use such terms as brethren (Acts 15:1; 1 Corinthians 16:20, NAS), disciples (Acts 11:26; 14:24, NKJV), and saints (Acts 9:13; 2 Corinthians 13:13, ESV). Before his conversion, Saul of Tarsus sought out those “who belonged to the Way” (Acts 9:2), indicating that an early label for Christians could have been “people of the Way” (see also Acts 19:9; 24:22).
Believers in Christ came to be called “Christians” during a time of rapid expansion in the church. Persecution had forced many believers from Jerusalem, and they scattered to various areas, taking the gospel with them. The evangelism was at first limited to Jewish populations. That changed when “men from Cyprus and Cyrene, went to Antioch and began to speak to Greeks also, telling them the good news about the Lord Jesus. The Lord’s hand was with them, and a great number of people believed and turned to the Lord” (Acts 11:20–21). Barnabas was there in Antioch, as was the newly converted Saul, and they were both teaching in the church. “And in Antioch the disciples were first called Christians” (Acts 11:26, BLB).
At the time that believers got the appellation Christians, it was common for the Greeks to give satirical nicknames to particular groups. So those loyal to the Roman General Pompey were dubbed “Pompeians,” and the followers of General Sulla were called “Sullanians.” Those who publicly and enthusiastically praised the emperor Nero Augustus received the name Augustinians, meaning “of the party of Augustus.” To the Greeks, it was all a fun word game and a verbally dismissive gesture. Then a new group cropped up in Antioch; since they were characterized by behavior and speech centered on Christ, the Greeks called them “Christians,” or “those of the party of Christ.” "
