Greetings James:
Our oak trees here in the Midwest tend to drop their acorns beginning in August and likely continue through October. Variability in acorn drop is due to species, age, genetics, environmental factors, and of course latitudinal gradients.
Illinois and Florida share some common oak species; however, there are a fair amount of oaks native to Florida that are not found here in Illinois (e.g., live oak, laurel oak, myrtle oak, etc.).
For a more state-specific answer, please try contacting a state extension forestry specialist at the University of Florida :-)
