I'll second Tosca in Hingham. Of anyplace I can think of right now that has live jazz/piano, they have the best food.
The Red Parrot in Hull has live bands on weekends during the summer...and previous summers they had jazz brunch on Sundays, not sure if that is still going on this year. The food at The Red Parrot is nothing special.
Its not exactly the type of music you were asking for, but sometimes on weekend nights El Sarape in Braintree has a musician or 2 squeezed into the corner of the bar area playing guitar and singing.
5 Commercial St, Braintree, MA 02184
14 North Street, Hingham, MA 02043
The Snug in Hingham has a variety I think on Saturdays. The Next Page in Weymouth has Blues on Thursdays. The Varsity Club in Quincy has music on Fridays and Saturdays. Don't expect too much on the food end, and I'd call first to get the program.
The Next Page Cafe
550 Broad St, Weymouth, MA 02189
33 Independence Ave, Quincy, MA 02169
is your emphasis on the music or the food? most times you sacrifice one for the other. Anyway, in Marshfield there is Poopsies on 139, pizza and beer type place with usually asoloist doing current stuff, there is the Fairview Inn on the water, very laid back atmosphere, moslty 30/40/50 something couples small groups. A new place, The Bailey (spelling may be wrong) Irish pub feel with usually Irish trending music. The Deck also has music, mixed crowd but usually a fun time. Depending on what you want for food, the Fairview is probably the most upscale, then the bailey, Then the Deck and finally Poopsies is more of a pub
243 Church St Ste 1, Pembroke, MA 02359
212 Prentice St, Holliston, MA 01746
Fairview Inn and Restaurant
133 Old Ocean St, Marshfield, MA 02050