There's no truth to it. While there were some proposals for a "demonstration" of the atomic bomb prior to use, there's no evidence that any of them were discussed with Truman. These were discussed and rejected at a lower level (the Interim Committee). The number of people who had access to Truman and also knew about the atomic bomb was very small. While there were a few in that group who later said they doubted the wisdom of dropping it (including Dwight D. Eisenhower), it is fairly clear that none expressed their doubts to him. And it is also fairly clear that Truman never had any kind of confrontation with anyone about its use (prior to its use).

Whomever said this story might have been getting it convinced with the one time Truman met J. Robert Oppenheimer, after the war. The latter came to his office, expressed that the atomic bomb had been a heavy burden for him, and Truman got mad that Oppenheimer claimed to feel the burden that Truman apparently felt was reserved for himself. He supposedly later told his staff to never let that "crybaby" back into his office.