Hmm you could just observe first and see if your baby can improve on his/her own. I would suggest monitoring in intervals of 7 days, to see PD if no improvement or gets worse. From my personal experience, it was hard for my baby to get well without medicine. My baby had a cough at 3mo and she caught a new cold at the tail end of the first one so it dragged for 7-8 weeks. Young babies tend to take a while to recover, so even when they are 'well', it can take a couple of weeks for the symptoms to clear up. Every baby is different though, so keep monitoring to see if your baby's feeding and sleep are getting affected by the cough. Should see PD when they are being affected.
Don't do any medication without pedias advice you better bring your baby to a pediatrician
Shivashankari Rajamogan