I have a 22 hp zero-turn mower. How can I find out if the mower is charging the battery?

Hi,
Measure DC volts of battery with engine off - should generally be between 12.6 - 13.2 V for a healthy battery. Then crank engine and throttle to full speed and measure again. The reading should be somewhere around 14.5 V, results could vary especially just after cranking because the battery should "want" even more because of the current flow out to crank the engine. Best to wait 3 - 5 minutes until it stabilizes to measure.
The first measurement was done because most charging systems are dependent on a healthy battery as a reference for the regulator.
Peace,
Clarke

With a good digital meter with a mV scale, you can measure across the cable to the battery at a high RPM. This generally will indicate current flow. If you do a reference value when things are good, it will help you later to troubleshoot problems.