Can a Bump Emerge a Few Hours After a Botox Injection in the Crows Feet Area?

it appears that you may have a small hematoma. A bruise of the skin with a small collection of blood cells that leaked from a capillary during the sudden intense pressure of the cough.  There are numerous capillaries and veins near the eyelid and if there was minimal perturbation of a vessel with the insertion of the tiny needle of the botox the cough could have opened a slit leaking out blood cells.  Usually it seals itself and the hematoma resorbs. Ask your doctor what they would recommend you do to expedite the resolution of this. Light compression is recommended usually for twenty minutes if this occurs, as soon as it occurs.  Some patients take blood thinners that increase the risk of this happening. It's not the botox but possibly the injection and it can happen to any patient and any physician. You should see your doctor for followup, but contact them now to discuss your current regimen.

the bumps that appear after the Botox injections are a result of minor vessel injury that can cause localized bruising. bruises are not permanent. they go away in 7-10 days. icing the areas helps. these can be covered with makeup as well.

That is called a mouse.  You got yourself a nice little bruise there.  It will eventually go away but it takes time.