I think that one of the fundamental internal conflicts that is present in Fitzgerald's work is how material success and happiness do not always accompany one another. Dexter is the embodiment of the American Dream. He works hard in pursuit of "making it" from a material point of view. He succeeds in this and profits in both name/ reputation and monetary value. Yet, despite this, he is still lacking happiness. It is here where an internal conflict is demonstrated in a strong and intense manner. His pursuit of Judy and the frustration that results in this is reflective of this. Dexter has a difficult time fully grasping how to reconcile happiness and success. Even when he should be relatively content with Irene, the need to have more and to "have it all" is present when he leaves her upon Judy's fleeting presence in his life. The ending, one in which success has visited Dexter and some level of content in the emotional realm, is also demonstrative of this internal conflict. When he finds out how awful Judy's predicament is, Dexter can only weep because "that thing is gone." The dream of happiness is for what he weeps, demonstrating that happiness and success are not immediately automatic.
The internal conflict is that Dexter struggles with himself to be rich and successful for Judy.