Even though that's the joke, that Greeks eat lamb and don't understand vegetarianism, Greeks don't traditionally eat a ton of meat in Greece. It's just that, Greeks don't really understand a diet where anything is restricted. Greeks don't overthink food. They just want it to be delicious, natural, and fresh. Food is often plant based, with meat reserved for special occasions or once in a while. In the coastal areas they also eat a lot of seafood since it is so plentiful.
There are so many vegetarian Greek recipes! My favorites are briam, bamies, fasolia, and xorta. There are so many options! Especially in home cooked food. Taverna or restaurant food is a little more difficult.