Your question uses an incorrect transliteration of the name of God. The Tetragrammaton would be more correctly written in English as YHWH. There is no 'J' in Hebrew, although there is in standard Arabic. I find the use of a letter whose sound is found in Arabic and not Hebrew rather ironic given the protestations to this point. The word you write for the name of God only appears seven times in the KJV Old Testament, even though YHWH is found 6807 times in the Hebrew Bible. It has disappeared from versions like the RSV, NKJV, NIV, NASB, and ESV. The word for the name YHWH is not found in the New Testament. I will correct your question and answer it for you.
Is the God of Abraham, Isaac, and Jacob the same in the OT and NT ?
Yes