I've had two cats with allergies that became asthma as they aged. If it's that "old man" cough, that's probably what it is; allergies are horrid this year.
As long as your cat is otherwise normal, I wouldn't freak out. Wait a couple days and see if symptoms improve. If you have a HEPA vacuum, try that. Wash the cat's bedding, vacuum any perches etc. If your cat is still willing to eat, drink, use the litter box, play, etc. it's probably nothing to worry about.
If the cat seems lethargic, won't eat, isn't using the litterbox, won't come out from under the bed even for a favorite toy/treat... then you need to get a vet involved.