Maybe. Facial flushing may occur on consuming food one may be allergic to. Consuming food that immune system considers to be harmful, it fights off the foreign substance. As a result, cutaneous mast cells release histamines into the blood. This causes skin response and other allergy symptoms. Facial flushing might develop due to the release of histamines into the blood.
Unlikely. Your baby may be having flushing of skin because of changes in blood flow which happens in babies, it is due to instability of blood vessels.This condition goes away.But it is better ,consult your doctor , who will examine your baby and explain to you if it allergy or something else.