What was the relationship between Cornelius and Jesus?
Peter told him that Jesus went about doing good. Cornelius and his household received the Holy Spirit and were baptized in the name of Jesus Christ. Acts 10:38, 44-48
What the Bible says
Acts 10:44
While Peter was yet speaking these words, the Holy Ghost fell on all them that heard the word.
Acts 10:48
And he commanded them to be baptized in the name of the Lord Jesus Christ. Then they desired him to tarry with them some days.
Douay-Rheims Bible (Challoner revision), public domain.
What the Church teaches
The Catechism recalls that the apostles baptized Jews, God-fearers, and pagans.
Catechism 1226 · Catechism of the Catholic Church