This is a a hard concept to visualize. We discover more about who God is as we look at the whole Bible. We may not fully grasp everything God reveals about Himself, but nevertheless, our inabilities doesn’t limit the truth of who he is.
In answer to your question we need to identify what parts of the Bible declares that there is one God, and where it states that the Father is God, the Son is God and the Holy Spirit is God.
And remember, we have to come to grips that the Bible does not contradict itself.
Here we go:
1) The Bible has always said there is one God:
"Hear O’ Israel the Lord our God, the Lord is one." (Deut 6:4)
Jesus even confirms this in all that he says and teaches (Mk 12:29-30).
2) However, the Bible says:
The Father is God. (Matthew 6:8, 7:21, Galatians 1:1)
Jesus is God from the beginning (John 1: 1-18)
The Holy Spirit is God (John 15:26, Mark 3:29, 1 Corinthians 6:19).
3) There are occasions when God refers to himself in plural terms (Genesis 1:26, 3:22, 11:7, Isaiah 6:8)
We cannot conclude from this that the Father, Son and Holy Spirit are independent Gods, because the Bible says there is ONE God.
Additionally, if we conclude that they are three independent Gods then we have broken the first of the 10 commandments in the Bible -
"You shall have no other gods besides me." (Deut 5:7).
So, it stands to reason that they cannot be independent.
How do we know this? Jesus explains it in John 14 - 17
To summarize Jesus says"I and the Father are one" (John 10:38, 17:11,21)
"I am in the Father and the Father is in me" (John 14:11) .
This means Jesus and the Father must be the same God but distinct. Now let’s consider what Jesus says about the Holy Spirit:
"I will send the Spirit of truth" for he lives with and will be in you" I am in my Father, and you are in me, and I am in you". (John 14:17-18, 20)
Now we have to solve this riddle:
How can the Father and Son be within each other at once? How can the Son and the Holy Spirit be in me at once? The answer is: Only if they are one and the same yet distinct persons.
And it stands to reason as we put together these facts that they are the same God, but different persons.
Can I reassure with the following: it doesn’t matter if we can’t imagine how this works in our minds! It is perplexing because there is nothing we have seen physically in our lives that can accurately illustrate this mystery. However, it is important to realize it is an eternal truth which the Bible has revealed to us! It’s not man made.