This is a straightforward name of the Holy Spirit, the third person of the Trinity. The name occurs several times in the Old Testament but just twice in the New Testament. The most common way to speak of Him is as the Holy Spirit. Jesus told us that God is spirit and we must worship Him in spirit and in truth.
Spirit of God in Hebrew:
Strong’s Concordance numbers: 7307, 430
Bible references: Gen. 1:2; Ex. 31:3; 35:31; Num. 24:2; 1 Sam. 10:10; 11:6; 19:20, 23; 2 Chron. 15:1; 24:20; Ezek. 11:24.
And the earth was without form, and void; and darkness was upon the face of the deep. And the Spirit of God moved upon the face of the waters. (Genesis 1:2 KJV)
Strong’s Concordance numbers: 7307, 410
Bible reference: Job 33:4
The Spirit of God has made me,
And the breath of the Almighty gives me life. (Job 33:4 KJV)
Spirit of God in Greek:
To pneuma tou Theos
Strong’s Concordance numbers: 4151, 2316
Bible references: Matt. 3:16; Rom. 8:9. Twelve times in total, including “Holy Spirit of God” in Ephesians 4:30.
And Jesus, when he was baptized, went up straightway out of the water: and, lo, the heavens were opened unto him, and he saw the Spirit of God descending like a dove, and lighting upon him. (Matthew 3:16)