What are the believers in the zuboor called?

From Surat al-Isra (Quran 17:55)

And your Lord is most knowing of whoever is in the heavens and the earth. And We have made some of the prophets exceed others, and to David We gave the Zabur.

The Psalms of David from the Old Testament, normally considered to be the current form of the Zabur, are accepted as scripture to the Jews. Therefore, it is reasonable that any followers of the Zabur would be called Jews, as they were the same Jews who originally followed the Torah. There is no reason to believe they would become a distinct religious identity founded around their new scripture, as the Christians later did.

There is some speculation that the Sabians referred to in the Qur'an (e.g., Quran 5:69) were a people who followed the Zabur in particular. This claim is somewhat substantiated by the following hadith (excerpted from Sahih Bukhari 1:7:340):

Abu 'Abdultah said: The word Saba'a means "The one who has deserted his old religion and embraced a new religion." Abul 'Ailya said, "The Sabis are a sect of people of the Scripture who recite the Book of Psalms."

However, the exact identity of the Sabians, or even the exact nature of the Zabur itself, is not something that is agreed upon by the scholars.