This is a rather tenuous conceptual connection but it is not without some potential.
The way that I would draw a connection in this would be to first attempt to prove that the Great Depression was caused, at least in part, by wealthy people (bank owners and financiers) acting without regard for morality.
There is evidence that the people in a position to control the banking system pulled out all their own money from the stock market and from the banks before the big crash. The story goes that they didn't warn anyone else.
This kind of behavior from those in a position of privilege can be connected to the behavior of the characters in Fitzgerald's novel. It is not the people of privilege who suffer from their own behavior, but others who do the suffering for them. This is arguably true of the Great Depression and certainly true of The Great Gatsbyas Myrtle and George Wilson both die in the end, while Tom and Daisy walk away, able to leave their house and simply buy another one away from the turmoil they have been instrumental in creating.
Clearly, this is a conceptual connection between the novel and the economic downturn, open to debate.