For Greece, the reasons for conflict are the same that used to be in the battle of thermopylae? Is it the same ancient culture conflict still responsible for these wars?
Hardly. Turks arrived in Turkey a millenium or more after Thermopylae, and to say that Greek culture of today is "the same" as that of the group of people who lived there in the Bronze Age is quite obviously false.
No, it’s a populist nationalist government in Turkey that’s continuing to destabilise the region. Greece has no desire for conflict, but if they are invaded, which is what one nation’s military entering the territory of another nation uninvited is traditionally called, they are within their rights (and duties as a NATO member) to respond.
More important is that Greece is part of the EU while Turkey is not, and EU members are supposed to come to the aid of any member that is the victim of military aggression.
Many Ukrainians regretted not being part of the EU when Russia invaded.
Not so true about EU, as EU has no common foreign policy or army. On the other hand if a NATO member gets attacked its part of the treaty that others come to help, the problem in this case is that the attacker is also a member and I don't think it's clear what happens in that case.
NATO is more and more an empty shell organization. The real question is: will other European nations side with Greece and help her militarily in case of an armed conflict? I’m particularly concerned with the position of France, which has one of the strongest EU military and possess an aircraft carrier. The political leaders have no balls however and they could realistically let Turkey invades part of Greece without moving a finger. The second issue is Germany position, which not only has historically been allied with Turkey, but also hosts a huge Turkish diaspora, and could political veto EU decisions.