What is the correct time for Isha prayer in Vienna, and is it permissible to pray in the Turkish mosque which calls for Isha prayer at 22:10, while the Islamic Center calls the adhan at 22:29?

The legal time for Isha prayer begins at the disappearance of the red twilight. Whichever of the two times (mentioned in the question, though not provided in the snippet) coincides with this timing is considered legally valid. Prayer before its legal time is impermissible and invalid. If it is unknown which of the two times coincides with the legal time, then investigation is obligatory. If that is not possible, then the later of the two times should be adopted as a precautionary measure.