Absolutely not. We do not see a breakdown of the relationship between the Father and the Son in the sense that the Trinity is somehow breaking apart. That is not what is going on.
The language of separation that is used, “My God, My God, why have You forsaken Me,” describes being under the wrath of God and being separated from the presence of His graciousness. Ultimately, there is nowhere to escape from the presence of God for anyone who is under His wrath.
Something quite unique happens at the cross—the eternally beloved One has the wrath of God poured out on Him. He’s never known that before, but He remains the eternal Son. He’s not suddenly split apart from His Father. It is an utterly unique moment in their relationship, but He still remains the eternal Son.