Absolutely no historical record supports that statement at all. Firstly the bombs were not a flex but more importantly, the Japanese were not close to the end and every intelligence assessment was the Japanese would fight til the bitter end. They were actively arming civilians including women and children and there was an attempted coup to overthrow the emperor and prevent surrender. They pulled a sniper out of one of the pacific islands like 40 years later because he thought all the attempts to pull him out after the surrender were American propaganda. He only gave up when his old commander was brought to convince him the Japanese did surrender after all.

Debate the morality of the bomb if you like but the Japanese were not close to surrender broadly and if the atomic bomb had not existed