Am I having asthma symptoms if I just cough when I'm mowing the yard?

Grass allergy. Could be exertional asthma or asthma triggered by cut grass. One does not need to wheeze to have asthma.
Lots of possibilitie. Asthma is a chronic inflammatory condition of the lungs. Typical symptoms include cough, wheezing, shortness of breath to name a few. Cough can have multiple causes including asthma, post nasal drip and reflux disease to name the most common. But other more very serious disease can cause cough as well like infection or cancer.
Possibly... Your cough may represent bronchospasm, with the grass serving as a trigger. See your doctor so this diagnosis can be made and treatment rendered.