How can I find out if my surge protector is putting out bad (dirty) power?

Hope that Tony stops by, the perfect guy for your concerns. Meanwhile, sans surges, you could still have an odd voltage issue. In my younger days, for expensive (rack audio) equipment, a line power "conditioner" was appropriate, the same function now in (AVR) better UPS units primarily intended to keep you powered up to save work in progress. Using a multimeter would at least get you a quick look at what's available at a given moment. And I'm not super confident in the quality and responsiveness of your typical Chinese surge protectors, either (mine are Belden -who for all I know are now also Chinese - sigh). Good luck, and wait for somebody better than me here - there are a bunch.

It would not be because of the surge protector but from the power company that is connected to the house. However you can replace the surge protector with one that protects against dirty power but these will be expensive.

Your surge protector can only put out what the power company is sending to your home. It cannot change anything.

You will need a multimeter that measures hertz. In North America, the meter should read around 60 hertz. If you have dirty power, the meter will fluctuate.