Step tracking while riding on a mower

Yes if will..it happens to me. You could use battery saver mode and have it stop tracking activity.

Try this:  Check the step count on your watch -- get on the mower for a few minutes --  see if the step count increased.  That'll give you an answer. 

Step counting with a wrist-worn accelerometer is always going to be a rough estimate.  Any movement or impact it senses could possibly be counted as a step.