Why did the Greeks, carrying out an active sea trade, do not have a navy?

Because Greece rarely waged wars with other states, most often the Greek policies were at enmity with each other, but basically all the clashes took place on land. Only after Persia started a war against Greece, the Greeks were forced to create their own fleet. Thus, already in 480 BC. the Greek fleet under the command of Themistocles was able to defeat the Persian armada in the Battle of Salamis.