"Church of God" is used many times to identify God's church in the NT. Is this what God's church is and should be called/ named today?

Yes. In fact, the Church of God is the ONLY true church today.

When I search the NASB and the KJV for the expression "Church of God"

It only comes up as "church of God"

In 2 Thessalonians 1:4 it reads 'churches of God' "So that we ourselves glory in you in the churches of God"

Thus in God's word, this expression is NEVER used as a title.

A better translation of the Greek is 'congregation' meaning the people and not the building.

Thus the congregation of God and the church of God are really the 'people of God'

Who are called 'Christians'

Who are called 'Witnesses'

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In modern times, the phrase "Church of God" is already taken, and is used by a denomination. I will always favor what Acts 11:26 declares. The followers of Jesus' teachings were first called "Christians" in Antioch, Syria, about 43 A.D.

God's church is hidden, so how can you name it?

Actually that is not given as a name for the church, but the true church is the Church of God. True Christians belong to God.

Jesus taught his followers only one religion, the true religion. It is like a road leading to everlasting life. Jesus said: “Few are finding” that road. (Matthew 7:14) God accepts only worship based on his Word of truth. All true worshippers are united in one faith.​—Read John 4:23, 24; 14:6; Ephesians 4:4, 5.

The source below shows identifying points of true worship declared by the Bible.

In the Old Testament God declares, "I am Jehovah and there is no other."

The Old Testament is a lot better.

Seems possible, then there's the reference to God's church as Nazarenes.