"There Is One God And One Mediator Between God And Men, The Man Christ Jesus" 
1 Timothy (2:5)

Does the Bible call the Holy Ghost a second or third person in the Godhead?

No. The Holy Ghost is the one Spirit of God, the one God Himself at work in our lives (John 4:24; I Corinthians 3:16-17; 6:19; 12:13).