How can I tell if the ionizer on my air filter creates ozone?

That’s a good question! HEPA filters generally do not rely on external power to ionize the air, and should produce no measurable ozone. Some electrostatic precipitators (and UV germicidal lamps that were used in smoke filters) do make ozone and possibly some nitrogen oxides, which not only damages textiles and rubber products, but even the lungs with extended use.
If there is a metal frame or pins with high voltage across it, it’s suspect, but if it has an activated carbon filter afterwards, not much ozone should come through. If there is an on/off switch for the ionizer, just leave it off.
There are a number of ozone test kits, from a few dollars for one-time paper test strips (probably what you might use) to machines costing hundreds of dollars.