Will the mark left after Botox fade?

It is very common to hit little capillaries and veins (particularly those around the eyes) with Botox injections. Some patients even may bruise as well. Both normal and common. Generally these little broken capillaries and/or bruises absorb within 1-2 weeks. If that little red mark remains visible after a few weeks, it can be treated with IPL or BBL therapy.

The red marks may be bruising or where the needle was inserted. Either way I would not worry about it, unless it starts expanding. 

It's not unusual for a bit of blood to get trapped under the skin or for patients to be left with a slight red mark following any injectable. Your injector is correct—this should fade in about a week or so. Keep an eye on the area and contact your injector if it seems to be getting worse instead of improving.