There is no scripture that any apostle ever baptized in the name of the Father, the Son, and the Holy Ghost. Baptism was only in Jesus’ name during the first century. Should we only baptize in Jesus’ name?
Let’s read Matthew 28:18-20 and Acts 2:36-38 and see if there is anything said there that could help answer your question.
“And Jesus came and spoke to them, saying, ‘All authority has been given to Me in heaven and on earth. Go therefore and make disciples of ALL THE NATIONS, baptizing them in the name of the Father and of the Son and of the Holy Spirit, teaching THEM to observe all things that I have commanded you; and lo, I am with you always, even to the end of the age” (Matthew 28:18-20).
“’Therefore let all THE HOUSE OF ISRAEL know assuredly that God has made THIS JESUS, WHOM YOU CRUCFIED, both Lord and Christ.’ Now when they heard this, they were cut to the heart, and said to Peter and the rest of the apostles, ‘Men and brethren, what do WE DO?’ Then Peter said to them, ‘Repent, and LET EVERY ONE OF YOU be baptized in the name of Jesus Christ for the remission of sins; and you will receive the gift of the Holy Spirit’” (Acts 2:36-38).
I have capitalized the words that may shed light on the question. The baptismal formula that Jesus gave was for the world of GENTILE NATIONS who were steeped in idolatry and were thus ignorant of the Triune God (the Father, the Son, and the Holy Spirit). We read in Ephesians 2:12 that they were “without hope and without God in the world.” In preaching to them the disciples would have to teach them the truth of God. We see a prime example of that in Acts 17:22-31 where Paul saw an inscription on one of their idols which read, “TO THE UNKNOWN GOD.” He then said to the idol worshipers in Athens, “Therefore, the One whom you worship without knowing, Him I proclaim to you” (verse 23) and he proceeded to teach them that God was their Creator and that one day He would judge the world by His Son, “the Man whom He has ordained” (verses 24-31). We read that “some…believed” (verse 34) and when they were baptized, they were no doubt baptized “in the Name of the Father, the Son, and the Holy Spirit.” By doing so, they were identifying with the Triune God and confessing Jesus as their Lord.
On the Day of Pentecost, the Apostle Peter preached to “the house of Israel” who had crucified Jesus Christ. He goes on to inform them that God had made Him both LORD and CHRIST. When they heard this, they were “cut to the heart” (convicted of having Jesus crucified) and wondered what they should do. Peter tells them to REPENT of this sin and to “be baptized in the name of Jesus Christ.” They had been guilty of having Him crucified because He claimed to be the Son of God (see Matthew 27:57-68). Now they learn that He is truly the Lord of glory and to prove they were repenting of having Him crucified, they were willing to confess Him as their Lord by being “baptized in His name.” In their baptism they were also confessing He was indeed the Son of God (part of the Triune God).
We won’t take the time to look at the other accounts where people were “baptized in Jesus’ name” (the Samaritans in Acts 8:12, 16; Cornelius and his household in Acts 10:1-2, 34-48; and disciples of John the Baptist in Ephesus in Acts 19:1-5), but in each case they all had a knowledge of God but were ignorant of Jesus Christ. They were all taught (by Philip in Acts 8; by Peter in Acts 10; and by Paul in Acts 19) the truth about Jesus Christ as the Son of God who died for them on the cross and after believing they too were “baptized in the name of the Lord Jesus.” In doing so they confessed Him as their Lord and Savior. (DO) (711.3)