The majority of asthma cases are diagnosed in childhood or in the early teenage years. When one develops new onset asthma and a cough at a late age, this is a cause for concern.
It is unlikely that one can have new onset asthma unexpectedly in the later years of life if the past medical history has been uneventful. Moreover, if you do not smoke and have not had any chemical related work history, then you definitely need more work up.
The first thing to do is ask your doctor to get a chest x-ray and a referral to a lung specialist. Asthma may be related to the heart but this can easily be ruled out by obtaining an x- ray. There are many other causes of the lung that can mimic asthma and most of them are serious. If your doctor does not intend to work you up more thoroughly, you need a new doctor.