Is it true that the Messenger (peace and blessings of Allah be upon him) mentioned that the Hour would be established after the demise of Baghdad and Damascus by name, and is this among the minor signs of the Hour, or is it a fabricated hadith?

No explicit prophetic tradition states that the end of Baghdad or Damascus is among the signs of the Hour.

However, there are traditions indicating: That Iraq, Syria (Al-Sham), and Egypt will withhold their dirham (currency) and qafiz (measure of grain). Imam An-Nawawi explained this as being due to the conquest of these lands by non-Arabs and Byzantines. That the Romans will encamp in al-A'maq or Dabiq in Syria before the Hour, then fight the Muslims, leading to the Muslims' victory and the conquest of Constantinople.