The answer is Yes. We know this because this is the emphatic declaration of the Bible (Mal. 3:6; Jam. 1:17; Heb. 1:12; 13:8). This is called the immutability of God, which applies to Christ as the Son of God, or God the second Person of the Trinity. This means the God of the Old Testament is the God of the New Testament. The Bible is a book of progressive revelation. What is seen in the Old Testament is true and accurate, but in the unfolding of Himself and His plan for man God gradually revealed more and more of Himself and His plan of salvation. Many think that God seems wrathful and harsh in the Old Testament as compared with the New, but a careful reading of both reveal God’s love, grace, mercy, and His judgment in both the Old and New Testaments. The grace of God in Christ was anticipated and prophesied in the Old Testament.
This does not mean that God’s way of dealing with man and revealing Himself has not changed. This is evident in the various dispensations or economies of God throughout history that are clearly evident in the Bible—the garden of Eden, after the fall, the time of Noah, the patriarchs, the time of Israel as a Nation, and now the church. In each of these there were different stewards of God, different responsibilities, failures, and purposes. But in all them after the fall, there was the hope of a coming Savior, the Messiah and the restoration of what Adam lost in the fall.
As such, we see a definite difference in the ministry of the Holy Spirit. For detail on this, please see the study, ABCs of Christian Growth, Part Two, lessons 4-5. In fact, I would encourage you to begin a study of the entire ABC series. This will answer tons of questions. You might want to follow this with a study on the Doctrine of God.