OP, go to your spin class and find the brand name and model number of their pedals. Then go to your bike shop (or one you can trust) or go online and get the appropriate cleats. Don't waste your time getting close. Cleats cost $15. Your shoes aren't the issue. Once you get the cleats sorted out, you'll be fine and glad you did.

There are lots of places on bikes where other manufacturers parts work just fine, sometimes better. But for the small steel SPD and SPD-like cleats and pedals, this is not the case. I pulled my foot out and crashed my first try with clipless using a mismatch very similar to yours 17 years ago. I now use all-Shimano SPD cleats and pedals on one bike. Work like a charm.

The much bigger plastic cleats of the old LOOKs are nowhere near as critical. I happily use black LOOK or Wellgo cleats on Forte Pedals with no issues. (The red cleats with float may be more critical, I wouldn't know. The float would kill my knees.)

Ben