The answers for the federal and state fiduciary returns are separate for TurboTax Business:
First, yes, TurboTax Business supports federal form 1041 and it supports the ability to e-file.
Regarding the state fiduciary return, you will need to check this link to see if the state return you need is supported in TurboTax Business. Some states are not supported due to low volume of requests.
https://ttlc.intuit.com/replies/3301325
