What kind of palm trees exactly? The fan palm grows wild in central Italy, although it's like a little stubby dwarf palm. Otherwise the date palm was known in Italy and Greece, brought from the east or from Africa, although since it doesn't produce edible fruit that far north it was purely decorative. This decorative use is pretty well-attested, as is its ritual importance. Originally the use of the palm frond as a symbol of victory was Greek, and Livy says that victors at the ludi Romani were awarded palm fronds in Greek fashion by the early 3rd Century. Date palms had religious significance at the sanctuary of Apollo at Delphi, and Pausanias mentions a grove of date palms at the sanctuary of Artemis at Aulis. Pliny mentions 49 kinds of palm tree, and they were not an uncommon part of the gardens of many villas, though I don't think you'd see anything but the native fan palms hanging out in most parts of the city...