Asalaam alaikum
IN THE GLORIOUS QURAN "Tubba" is mentioned twice but in both the places no chronical is given thus scholars and commentators have discussed extensively regarding their origin.
As you have mentioned we can find referring to the people of Tubba in surah Qaf and the only other reference we can find is in surah ah dukhan. More specifically in verse number 37.
Are they better or the people of Tubba‘ and those before them? We destroyed them because they were indeed Mujrimûn (disbelievers, polytheists, sinners, criminals).(44:37)
AND IT'S IN THE TAFSIR of this verse that we find more understanding regarding the Tubba. Below is an extract from the tafsir of ibn Kathir of this verse and it sheds light on the origins of the Tubba.
They Tubba were Arab descendants of Qahtan, just as these people (Quraysh) were Arab descendants of Adnan. Among the people of Himyar -- who are also known as Saba -- when a man became their king, they called him Tubba, just as the title Chosroes was given to the king of Persia, Caesar to the king of the Romans, Firawn to the disbelieving ruler of Egypt, Negus to the king of Ethiopia, and so on among other nations
(I recommend you to read the full text of the tafsir of this verse)
ALSO I found corroborating evidence to this by Ibn Khaldūn as he mentioned them in his al-Kitābu l-ʻibār ("Book of Evidence"):
This can be illustrated by what happened among the nations. When the royal authority of Ad was wiped out, their brethren, the Thamud, took over. They were succeeded, in turn, by their brethren, the Amalekites. The Amalekites were succeeded by their brethren, the Himyar. The Himyar were succeeded by their brethren, the Tubba's, who belonged to the Himyar. They, likewise, were succeeded, by the Adhwa. Then, the Mudar came to power.
NOW THAT we have established an understanding of this people we must also know that ibn Kathir has concluded that Tubba here means an individual and it's a righteous king by the name Asad Abu Kurayb who ruled for 326 years before the advent of our beloved prophet ﷺ.
(detailed account of this can be found in the tafsir of surah Saba [34])
After mentioning a lengthy incident of this individual ibn Kathir writes,
This incident shows that the Tubba` mentioned here had himself embraced Islam, but his people later went astray. On both occasions the Qur'an has made reference to the 'people of Tubba, and not to Tubba' himself . This is supported by the narratives of Sayyidna Sahl Ibn Sa'd and Ibn Abbas ؓ which are recorded by Ibn Abi Hatim, Imam Ahmad, Tabarani and others that the Messenger of Allah ﷺ said: لا تسبّو تبّعافانّہ، قد کان اسلم "Do not revile Tubba', because he had embraced Islam."
(again I recommend and advice you to read the full text as it would give you more clarity regarding this matter)
REGARDING the part of the question about a prophet being sent to them or if he himself was a prophet, I have not come across a definitive conclusion to that and it requires further research and as of now, this is what I know.
“I do not know whether Tubba was a Prophet or not, and I do not know whether Dhu’l Qarnayn was a Prophet or not.”
[Al-Haakim and Al Bayhaqi. See Saheeh al-Jaami as-Sagheer, 5/121]
Reference
may Allah the mighty and sublime grant us mercy and guide us to the straight path.
Allah knows best
