Sicily
No, it was Phoenician.
Carthage was a trading and sea power. Rome was an upcoming land power. They were complementary in keeping peaceful trade until their interests clashed over Greek Sicily. This island at the southern tip of the Italian peninsula had Greek colonies allied with Carthage. This combination threatened any advance to Sicily by Rome.
Carthage is not an island, but is near the Mediterranean Sea.
The first of the three Punic Wars fought by Rome and Carthage was over the control of Sicily.
In 146 BCE after they destroyed Carthage and sold its inhabitants into slavery. It then established a colony on the site to settle retired soldiers.
Rome against Carthage, on the island of Sicily
The people were sold into slavery, and it was used to settle retired Roman soldiers.
In 265 BC, Carthage controlled the island of Corsica, located to the north of Sardinia. This strategic position allowed Carthage to extend its influence in the western Mediterranean. Corsica was significant for its resources and as a naval base for Carthaginian operations. The island's control was part of the larger tensions between Carthage and Rome that eventually led to the Punic Wars.
No Greek cities helped Carthage. The opposite was the case. The city-states of the Aetolian League became allies of the Romans and fought against Macedonia alongside them.
No island was found by Carthage and Rome. The first Punic War, the first of the tree wars fought between Rome and Carthage, was over the control of Sicily.
They were originally allies in controlling Greek penetration in the Western Mediterranean with complementary forces - Rome on land, Carthage on the sea. They came into conflict as Rome, having dominated Italy, sought to extend into Sicily, clashing with Carthage's foothold on the Island. This brought on the First Punic War; their second and third wars were a struggle for control of the Western Mediterranean.
Sicily