Can I take the Train in Fitchburg which is the Purple line to Norwood Hospital?

Norwood Hospital is in Norwood, so you have to go to South Station to catch the Franklin Line train

Norwood is on the Franklin Line. You can catch Franklin Line trains from South Station or Back Bay. If you're coming from Fitchburg you can take the Fitchburg Line to North Station, then the Orange Line to Back Bay, and catch your Franklin Line train from there.

Back bay