Does the Bible say that there is only one person in the Godhead?

No.

The Bible does not define the terminology of the Godhead but scripture lets us know that the Head (the authority) of Christ is God, as Christ is the head (the authority) of man (1 Cor 11:3). Christ was the chosen lamb (the chosen body) whom God anointed with His authority to administer His plan of salvation (John 12:49-50).