FCAT results in demographically similar areas of Duval County are about the same as they are in St. Johns County (and Clay). You can sort the FCAT scores here.http://schoolgrades.fldoe.org/default.asp
A cursory review shows that by this past year's data, schools of all levels in Mandarin, the Beaches, and a lot of the Southside score similarly to schools in St. Johns (or Clay). There are also others throughout the rest of the county (including the urban neighborhood of San Marco) that also score highly.
St. Johns does have more consistency across the board - they have no schools that scored lower than a "B", and all of those "B" schools appear to be in the St. Augustine area rather than Ponte Vedra or the Julington Creek-Fruit Cove area. Presumably St. Johns' success is affected by the fact that a lot of families move St. Johns specifically for the schools - like seeks like, and all that.
Not that the FCAT is a great measure (it takes in no consideration for specializations, honors programs, International Baccalaureate, etc.), but it is a measure.