no, if you get a felt backed carpet you dont need to put an underlay down. If you have hessian backed carpet you will need underlay. But on stairs you're better off getting hessian (jute) back and underlay since it wears better
My father was a carpet fitter for 30yrs and alsways put underlay beneath the carpet even if it was felt back - the carpet will last longer and it will deaden the noise of persons using the stairs.