sounds like exercise-induced asthma not an allergy
You should consult your chest specialist or doctor to rule out exercise induced asthma.
Do you have any other symptoms?
Are you on any medications currently?
Have you tried taking inhalaer medications for your complaints? Have they helped?
Do you have any other symptoms like breathlessness, reflux symptoms or heartburn or nasal symptoms like nasal discharge etc. Do you smoke?
You should try steam inhalation, warm saline gargles, saline nasal drops and oral antihistamine medications and see if it helps with your symptoms.
If the cough still persists, it would be best to consult your doctor and also get a chest xray done to rule out a chest infection.
Let us know about how you are doing and if you need any other information.