“Why does my baby have green eye discharge?”

Yes, I recommend taking your child to their doctor. - Dr. Andrews

Eye drainage comes from the tear ducts which are also connected to the nose. If the whites of the eyes are pink, this may be pink eye. If the whites of the eyes are white and there is a green runny nose the this may be a sinus infection. If the baby appears I'll the baby should be seen by pediatrician. Is the baby is happy and not ill appearing, then you can see if it resolves on its own with saline eye drops and saline nasal spray

Absolutely. Please see an board certified ophthalmologist, especially if they specialize in pediatrics.