Turkey and Greece are pictured at first as big enemies, but it's not fully true. Sure, they are rivals and there were a lot of wars between them, but after all we can see that they like each other and actually are pretty good friends.
It's especially shown in one of the episodes where both of them are going to the sauna and taking bath together, normally talking with each other. Normally in their own way, of course. They also say that they like their wars and think of each other as worthy opponents.
The person they quarrel the most about is Japan, because both of them want him to be their own friend. It's the only thing and there's not really shown (in the anime) that they ever fought with each other for any other reasons.
TurGreece is a rarepair within the Hetalia fandom.