The God of the Old Testament and the God of the New Testament are vastly different entities.
If one were to take away the biblical writings of the two, and focus solely on the actions/commandments of them, it would be extremely hard to assume that both deities are the same.
The God of the Old Testament is depicted as a staunch ruler that favors obedience and abiding laws over true morality. You see it in how he "tested" Abraham by commanding him to kill his son, even though human sacrifice is detestable to Him. He rewards the Levites for killing their own families in loyalty to Him, and tortures Job to prove how devout his follower can be.
However, the God of The New Testament is depicted as a loving father, seeking to save humanity from themselves, even sacrificing himself in the process. You see it in the story of Jesus, in his teachings to love one another, and how the Father in Heaven will accept anyone who seeks him.
For example:
John 3:16 states that "For God so loved the world that he gave his one and only Son, that whoever believes in him shall not perish but have eternal life. "
Yet in the Old Testament, God required the slaughter of countless men, women, and children, mass-killed his own people when they stepped out of line, and even drowned the entire human population.
In fact, God was going to wipe out the Israelites entirely, but Moses convinced Him not to. Not because he loved the Israelites, but because killing them all would make Him look bad.
To go even further, it was the common belief at that time that only Jews could be saved by God, which is why books such as Jonah were written.
There's no explanation for when or why God makes such a change, He didn't start loving everyone after he sent Jesus, he sent Jesus because he loved everyone. But God's actions in the old testament all but debunk this idea. And beyond salvation itself:
James 4:8 says: "Come near to God and he will come near to you. Wash your hands, you sinners, and purify your hearts, you double-minded.",
Which once again means anyone can come to God, yet in the old testament God executed Korah for believing it.
1 John 4:12 claims " No one has ever seen God; but if we love one another, God lives in us and his love is made complete in us."
But God appeared before several old testament figures, such as Adam and Eve, Moses, Solomon, and Jacob full on wrestled with Him.
In the Old Testament, God had a man put to death for doing even minor work on the Sabbath,
While Jesus heals a man and has him break this same law.
And one of the most notable changes, God commanded in Deuteronomy 19:21 states
" Show no pity: life for life, eye for eye, tooth for tooth, hand for hand, foot for foot. "
But Jesus said the complete opposite when he claimed:
You have heard that it was said, ‘Eye for eye, and tooth for tooth. ’But I tell you, do not resist an evil person. If anyone slaps you on the right cheek, turn to them the other cheek also.” – Matthew 5:38–39
The actions of these two God's are mutually exclusive. Unless there is a major change within the core deity itself. But due to God's depiction of being perfect and unchanging, is impossible.
I'd go as far as to say that Allah, another derivative of God who is more traditional "eye for an eye", is a closer rendition of the Hebrew God than the holy trinity is.