Most cases of an infant cough are not serious enough to warrant an ER visit and usually just waiting it out and using a humidifier in the infants room can help alleviate the infants cough.
If the infant begins coughing up blood or has trouble breathing then you need to take the infant to the ER for the coughing.
If the infant is having trouble swallowing or having shortness of breathe then the infant should be taken to the ER.
Also if the infant has a fever of above 100 F or the coughing doesn't stop on it's own and it's making the infant have a hard time to breathe.
A temperature of 104 F or above you should take your infant to the ER to get the temperature to go down.
If the infants fever is under 100 F then usually no medical attention is required.
But if the infants fever reaches 100 F or above and doesn't go down on it's own within a day then take the infant to the emergency room.
The flu can kill infants, kids and even adults so if the flu doesn't go down soon enough on it's own then take the infant to the emergency room or urgent care center just to be safe.
If the infant gets a temperature of 104F or above then take the infant to the emergency room immediately.
If your babies fever is not at 100 F then they are likely fine and don't need to go to the emergency room.
If you cannot get the babies temperature to go back down to normal then you should take the baby to the emergency room because the fever could cause brain damage to the baby if it is not controlled or doesn't go back to normal.
If the babies temperatures gets to 104 F or above then that is dangerous fever temperature for a baby and the baby needs to go to the emergency room for medical treatment.
But if the fever gets to 104 F and then returns to normal within the day then they are likely okay.
But I would take the baby to the emergency room if the babies temperature ever got above 100 F just to be safe because fevers and the flu affect babies in a more dangerous way than it does adults.