Thank you for question on the first church in the Bible, “What was the name of the first church in the Bible?”
A church is NOT a building or a location. A church is a called out group of believers in Jesus Christ who meet together to worship, to learn the Word, to be mutually encourage by fellowship, to serve the Lord, and to proclaim the Good News (Acts 2:38–47). It is organized group and yet a living entity, organism, of believers.
The first gathered group of believers were in Acts 2 in the Upper Room where the disciples were worshiping, learning the Word, fellowshipping together, in sharing their possession and later proclaiming the Word. Did the believers have a name for themselves? They were known or called themselves the followers/disciples of the Christ. It may be that others identified this group of followers as Christ followers just as there were followers of John the Baptist.
Thus the Upper Room was the location of the first church. With the outpouring of the Holy Spirit on believers and the convicting work of the Holy Spirit, the Upper Room could not accommodate all the believers. Thus the believers met in the Temple area. The gathering of the large numbers of believers in the Temple area could have been called The First Church of Jerusalem.
SUMMARY: A church is not a location or building. It is the people who met in a specific place with specific Christian activities. One can call the Upper Room as the First Church of the New Testament.
For more perspectives:
https://www.quora.com/What-was-the-name-of-the-first-church-in-the-Bible