Persistent (3 months) cough in 22 month old. Is it asthma?

Possibly. There are several things that can cause a persistant cough(asthma, reflux - heartburn and others). I presume your child is not in distress. And that he/she has been examined by your pediatrician..In that case i would recommend seeing an allergist..To take a full history(including family history of allergies/asthma) and examination. Several possible medications can be tried. That should be helpful.
Mgt. There are many possible causes of persistent cough at this age. A virtual appointment may be arranged, but asthma is one common cause.