Does he have some kind of syrup that I could give him to calm him down?

Yes, there is a type of syrup that could be given to help soothe the cough. Most cold medications are not recommended for children under the age of six, but there is a syrup called Hélixia that could be worth trying. It is made from ivy and has an expectorant effect, helping to clear the secretions that often cause coughing. It has a honey flavor that makes it easier to give to children. Coughing is usually one of the last symptoms to go away after a respiratory infection, so it's normal for it to persist for a while. If other symptoms appear or the cough lasts for more than a few weeks, it would be important to consult a doctor, but for now, it may not be necessary.