Nigel J says it well, they had the same doctrine but there is no way to know for sure if they are the same people. It's best to go with "I don't know if they were the same people or not" and focus on the main point of the story.
I don't mean to be overly detailed but it helps me to understand something if I look at the context.
Looking at the whole passage:
Acts 15
1 Certain people came down from Judea to Antioch and were teaching the believers: “Unless you are circumcised, according to the custom taught by Moses, you cannot be saved.” 2 This brought Paul and Barnabas into sharp dispute and debate with them. So Paul and Barnabas were appointed, along with some other believers, to go up to Jerusalem to see the apostles and elders about this question. 3 The church sent them on their way, and as they traveled through Phoenicia and Samaria, they told how the Gentiles had been converted. This news made all the believers very glad. 4 When they came to Jerusalem, they were welcomed by the church and the apostles and elders, to whom they reported everything God had done through them.
5 Then some of the believers who belonged to the party of the Pharisees stood up and said, “The Gentiles must be circumcised and required to keep the law of Moses.”
Time line:
Verse 1
- The scene is Antioch
- "Certain people came down from Judea to Antioch". We don't know who but we know they taught circumcision.
Verse 2
- Paul and Barnabas disagreed with the people who came from Judea.
- Paul and Barnabas are sent to Jerusalem do discuss the question of circumcision.
Verse 3
- Paul and Barnabas travel through Phoenicia and Samaria and report on Gentile conversion
Verse 4 the scene shifts to Jerusalem
- In Jerusalem they are welcomed and they give a status report
Verse 5
- "some of the believers who belonged to the party of the Pharisees" argue for circumcision.
We don't have enough information in this passage to say with any certainty if the "certain people" in verse 1 are the same people as the believers in verse 5. We do know that they held the same belief in circumcision and that drives the main topic of the passage.
The point of 15:1-6 is to introduce the question of circumcision and the larger critical question of works vs. faith and to set the stage for Peter's speech starting in verse 7.
The pivotal verse in the chapter is:
11 No! We believe it is through the grace of our Lord Jesus that we are saved, just as they are.”