There are two specific examples that allude to the dates consistent with the Great Depression. In chapter 1, Scout narrates:
Maycomb County had recently been told that it had nothing to fear but fear itself.
This alludes to the words of Franklin Delano Roosevelt who encouraged America that the country was going to make it through the Great Depression.
Later in the novel, Atticus uses a year in his closing argument that gives readers a date certain throughout which the entire novel took place.
There is a tendency in this year of grace, 1935, for certain people to use this phrase out of context, to satisfy all conditions. (Chapter 20)
These are the ways that we learn that it is The Great Depression through just the knowledge of time, but throughout the novel, Atticus in particular gives merit to the era because of discussion of money. He lets the kids know that they are poor... although not as poor as many others that they are around. He also refers to ways that doctors and lawyers are getting paid during this era: through donations of food and supplies.