Why does Nick go East and when does he do it?

We can find the answer to this question right at the beginning of the book when Nick Carraway is explaining who he is and the background to the current story. He explains that after his graduation from New Haven his career was then interrupted by the Great War, after which he felt very "restless" when he came back to the Midwest, which "now seemed like the ragged edge of the universe." For this reason, and to quell his own sense of restlessness (we can assume that Nick, like most returning soldiers, was probably suffering from shell shock on some level, and certainly from culture shock), Nick decided to go east and "learn the bond business."
Nick broached this idea to his family, having come up with it because many of his other single male friends were in the bond business, so it seemed like a good option. There is no suggestion that Nick was particularly drawn to it for any other reason, but his family approved of the idea, and so Nick did go east in 1922.