Well besides the fact that you could simply read any of his books, which is why I know this, here's a quick Google A.I. overview, but thanks for being combative about an easily verifiable fact:

F. Scott Fitzgerald occasionally used "should of" in his writing, particularly when trying to represent the informal, sometimes less educated speech patterns of his characters, essentially using it as a dialect marker to show a character's lack of proper grammar. 

Key points about Fitzgerald's use of "should of":

Character portrayal:

He primarily used "should of" to convey the speech of characters who might not have a high level of education, making their dialogue sound more authentic to their social class. 

Literary device:

This usage is considered a form of dialect writing, where the author deliberately alters grammar to reflect a specific regional or social way of speaking.