Whitman says yes and 32 years ago wrote,
Seeds of the South American sapote were introduced into South Florida in 1964. These had been obtained, at the request of the writer, through Lee Moore, from trees in the Amazon Basin at Iquitos, Peru. The resulting Florida-grown seedlings were later distributed among various members of the Rare Fruit Council International as a new introduction for further trial and observation. First fruiting occurred in 1973 on a tree grown in Miami by Bernard C. Bowker. A color photo of South American sapote fruit grown by the writer appears on the front cover of the above organization's 1976 Yearbook.
There are now several additional bearing specimens of the South American sapote in Dade County. One of these, in the yard of the writer, is 23 feet high with a 28-foot spread and a trunk diameter of 12 inches near ground level. Although Williams (3) writes of buttressed trunks, this tree is showing only a slight tendency in this direction. There are no limbs for the first eight feet, then branches appear in groups of five equidistantly spaced around the trunk in the same plane. These radiate out and ascend at varying angles of inclination from nearly horizontal to about 60 degrees. This branching pattern is repeated at four to five foot intervals with bare trunk in between. The tree has a spreading growth pattern with heavy, dense foliage dropping to within a foot or so of the ground. The leaves, up to 22 inches across, tend to thin out in winter. Their petiole is long, measuring about two-thirds the leaf's width.
In South Florida, the Quararibea cordata should be grown in full sun, and under favorable conditions, can increase in height at a rate of two feet or more per year. First fruit set for a young tree can be preceded by three or more unsuccessful annual flowerings. This bloom occurs during mid-winter with the resulting crop ripening the following November. Eight or more of the 4½-inch diameter sapotes can be clustered around a foot or less of branch, although usual fruiting patterns are more dispersed. The writer's tree, previously described, currently is carrying fifty-eight of the 'mango-melon'-flavored fruit.
Unquote.They range up to 5000ft altitude in their native distribution and vary much in quality,taste and size. Search some of the past pics I have posted of giant and low fibre lines.Anyway by selecting upland specimens with more cold tolerance further north in Florida should be ok for them.