No, there is no evidence for this idea.
Use of two names in Acts
The idea is perhaps based on a misreading of Acts 11:26 which suggests that the name was coined by the people of Antioch in response to the year-long preaching efforts there by Barnabas and Saul:
Acts 11:26 “and when he had found him, he brought him to Antioch. For a whole year they met with the church and taught a great many people. And in Antioch the disciples were first called Christians.”
This does suggest that the name arose among the people of the city, but there’s nothing in this verse that suggests that it was in anyway derogatory or that Barnabas and Paul did anything to discourage the group at Antioch being called “Christians”.
“For we have found this man a plague, one who stirs up riots among all the Jews throughout the world and is a ringleader of the sect of the Nazarenes.”
“And Agrippa said to Paul, “In a short time would you persuade me to be a Christian?” “
Peter: Glorify God in the name “Christian”
As Christianity expanded in the Roman empire it began to meet opposition not just from Jews but from pagans. This led on to the persecutions of Nero and other emperors, and various Roman texts and inscriptions contain attacks on “Christians” as a strange new sect. This then is Peter’s response:
1 Peter 4:16 Yet if anyone suffers as a Christian, let him not be ashamed, but let him glorify God in that name.