Can I give my 10 month old medicine?

So I think my son is getting sick. I’m not sure if it was the cold air or my nephew that is currently sick. Last night he kept waking up very fussy and when I would carry him I can hear him trying to swallow his saliva a lot and then he would cry. Today his boogers were running a little and it hasn’t happened for a few hours. I’m not sure if having the fan on also had something to do with it. Can I give him medicine or go to the doctor? Or?