Where can I find my monitor id number on my apple watch

What do you mean by "monitor ID"? Do you mean the serial number? If so, go to the Watch app on your phone. Then go to General>About>Serial Number.

If you mean something else, please explain further. What do you need this ID for?

Have you asked the people at the gym if the Apple Watch is a supported device?

You might try the BlueTooth ID. General>About>BlueTooth.