If we say "God the Father, God the Son, and God the Holy Spirit" can we say "God, God, and God"?

Is it correct to refer to God as "god, god, and god"?

That could be open to misinterpretation because there are not three separate gods in the Trinity but three Persons in one God.

Each Person of the Trinity is God (i.e., has the Divine Essence), which is God's existence. "God" refers to the Divine Essence; "Father," "Son," and "Holy Ghost" refer to the Persons.

On the unity of the Trinity, the Catechism of the Council of Trent says:

In the one Substance of the Divinity the Father is the First Person, who with His Only-begotten Son, and the Holy Ghost, is one God and one Lord, not in the singularity of one Person, but in the trinity of one Substance.* These Three Persons, since it would be impiety to assert that they are unlike or unequal in anything, are understood to be distinct only in their respective properties. For the Father is unbegotten, the Son begotten of the Father, and the Holy Ghost proceeds from both. Thus we acknowledge the Essence and the Substance of the Three Persons to be the same in such wise that we believe that in confessing the true and eternal God we are piously and religiously to adore distinction in the Persons, unity in the Essence, and equality in the Trinity.

Hence, when we say that the Father is the First Person, we are not to be understood to mean that in the Trinity there is anything first or last, greater or less. Let none of the faithful be guilty of such impiety, for the Christian religion proclaims the same eternity, the same majesty of glory in the Three Persons. But since the Father is the Beginning without a beginning, we truly and unhesitatingly affirm that He is the First Person, and as He is distinct from the Others by His peculiar relation of paternity, so of Him alone is it true that He begot the Son from eternity. For when in the Creed we pronounce together the words God and Father, it means that He was always both God and Father.**

^{…Dómine sancte, Pater omnípotens, ætérne Deus: Qui cum unigénito Fílio tuo et Spíritu Sancto unus es Deus, unus es Dóminus: non in unius singularitáte persónæ, sed in uníus Trinitáte substántiæ.
[…O holy Lord, Father almighty, everlasting God; Who, together with Thine only-begotten Son, and the Holy Ghost, art one God, one Lord: not in the oneness of a single Person, but in the Trinity of one substance.]}

Given that the Bible is perfectly clear that there is only one God, who alone is the true God, and that all other contenders for the title are false gods, shams, non-existent except in the minds of men, or (the one exception) the false god of this evil system, Satan the devil, the answer has to be "No, you cannot."

Nor does capitalising the 'G' or putting it in lower case as 'g' make any difference.

The God of Christians is a complex divine Being, not simple. There is no sinful human (i.e. all of us) who can comprehend the awesome Almighty God. Only what God has chosen to reveal of himself to us can inform our understanding. This is a matter of revelation, not theology. That is why, when the penny partially dropped with Peter, and he stated that Jesus is the Christ, the Son of the living God" - Jesus said God had revealed that astounding fact to him (Matthew 16:13-17). Unfortunately, Peter then went on to rebuke the Son of God for saying the cross, then resurrection, lay ahead. Jesus called Peter 'Satan'. What Peter said then was not of God and betrayed a very incomplete understanding of what being the Son of God the way Jesus was, signified.

After Jesus' resurrection Peter grasped that Jesus' power on earth was to bring in a heavenly kingdom in the world to come. To achieve that, Jesus had to suffer and die as a sinless sacrifice for sin. That was the way the Godhead had worked out - before creating anything - as the only, and the necessary way. The eternal Son had to lower himself for a season, to become a perfect human, conquer sin, death and the devil, then return in glory to be in the center of the Father's heavenly throne, surrounded by the seven-fold Spirit of God (Revelation 4:5 & 5:5-14 & 7:17 & 22:1). The Holy Spirit was equally involved in all of that as with creation. But nobody who thinks Jesus was created at any point in time (either as the first of God's creation, or at his conception in the virgin Mary's womb) ever gets it.

The three never work independently, but as one, in the Godhead. Total unity, total co-operation, even though humans can never see the invisible Father, only saw the Son as the man, Jesus, and only experience the power of the Holy Spirit who is invisible. Our understanding is limited due to our finite, sinful state. God is the Almighty and infinite, and chose to deal with our sin by sending the Son to save those who believe in him. To believe in Christ is to believe in the Father and in the Holy Spirit - not as three separate Gods, but as the one Godhead - complex.

This brings us to what you said, "it is said "God died for our sins" or "God suffered". Yes, in a sense that is true, though the Father never died - only the Son as a human died. But it was God, in Christ, suffering as the Son died for our sins. That is because the divine nature in Christ is the same divine nature as in the Father, and as in the Holy Spirit. Those who are called the early Church Fathers wrote about this. For example:

"Although [the Son] was God, He took flesh; and having been made man, He remained what He was, God... For we do not hold... that some part of the substance of God was converted into the Son, or that the Son was procreated by the Father from nonexistent substances, that is, from a substance outside Himself, so that there was a time when He did not exist." (Fundamental Doctrines by Origen, died circa 250, p1, pref.3-4, as in The Faith of the Early Fathers by W.A. Jurgens)

The Son, also called the Word of God, has always existed. For a short season he walked on earth as Jesus, to carry out the plan of salvation. But those who disagree with John 1:1-14 (and with Origen in that quote) think there was a time when he did not exist (unlike the Father, who is eternal). Well, Christ the Messiah has titles such as "The Everlasting Father, Mighty God, First & Last (= Alpha & Omega).

Some misunderstanding as to what the Greek word "only-begotten" means contributes to confusion, but there are already lots of answers on here about that. Just think on Jesus' words, "Whoever has seen me has seen the Father", and "I and the Father are one" and "that all may honour the Son just as they honour the Father. He who does not honour the Son does not honour the Father who sent him." To worship Christ is to worship the Father - not as a second deity, but as the one Creator God.

The Father and the Son share the one, divine nature, with absolute unity of the Spirit in that nature. That is why the answer has to be "No."

Language uses words with particular senses to talk about the world. It's a system of symbols which point to things in the world. One thing in the world, one referent, might have many different words (with many different senses) which can be used to refer to it. For example, one person could be referred to with the words "Mrs Smith", "Ruth", "mother", "grandma", "boss" etc. Though all of those words refer to the same person, they do not have the same senses and are not interchangeable.

Here's another example: at a meeting of world leaders the President of the United States, the President of Russia, and the President of Mongolia all had lunch together. In the context of each country's own media, it is common and appropriate to call each by simple title of "the President". But it would be very confusing to transform that statement into "at a meeting of world leaders the President, the President, and the President all had lunch together." And that's not just because they're all the same word; it would be wrong for US media to write "the President and the Chancellor of Germany had lunch together" if they were referring to the Russian president.

More is going on here than just confusing language however. The unqualified word "God" in Western Christianity (Catholicism and Protestanism) usually refers primarily to the Godhead, the singular essence of God, the one divine being in existence. My understanding is that this is one of the differences between Western and Eastern Christianity: instead of using "God" to refer to the godhead, Eastern Christianity usually uses it to refer to God the Father specifically.

When we say "Jesus the Son of God" we are referring to the second person of the trinity in particular. When we say "Jesus is God" we mean something different, that he is divine. But this does not mean that Jesus is the Godhead or the divine essence.

So if someone said "God, god, and god" Western Christians will only hear the first sense, that they are referring to the godhead three times, which means three divine beings, three divine essences, which is tritheism. Eastern Christians might instead understand it as referring to God the Father three times. But in neither case will it be understood as referring to the three persons of the Trinity.