How soon can you fertilize your lawn after spraying a weed killer (selective broadleaf)?

As soon as you want. Weed and feeds have broadleaf control mixed with fertilizer and are applied at the same time.

So if I've sprayed my St. Aug yard last Sunday with Image, I'm good to go with laying down a fertilizer?

A lot of my yard is brown now just due to the overwhelming amount of weeds.

Unless the label on the bottle of Image tells you otherwise, Sure you can apply it. Even though spaying herbicides for lawn weed control often stresses the grass to some degree, much, but not all, of the forms of nitrogen (N) in lawn fertilizers, for example urea, are relatively slow release forms of N, so your lawn is not going a big slug of N at the on-set anyway. That’s why you fertilize at 2 to 3 month intervals, these lawn fertilizers are designed to release their nutrients over a 2-3 period (speed of release is usually dictated by temperature, and to some extent precipitation).

You want to stimulate the growth of grass to fill in your areas were you killed weeds as soon as possible, otherwise the weeds will just return.

Of course water it in soon after (it will tell you that on the bag) or apply before an expected rain. Now of course I’m referring to lawn fertilizer only, not the weed & feed products.