Key West Harbor Inn's "Puerto Rico Room" is a great spot where if visiting and you need to breastfeed, you can go to feed your baby.

Nursing Mothers have breastfed their infants here and the area is very quiet and private, if that's what you prefer.

Key West is a very progressive laidback , but fun touristy place so public breastfeeding happens quite often here. 

The state of Florida also protects a mothers right to breastfeed in public so you can do so whenever and wherever you feel comfortable.