Could you give a brief summary of what the arguments are? (or links to studies making the arguments).
Al-Jallad thinks that the Quran could not have been written in Arabia Patrea. Per al-Jallad, even in the sixth century, the Quran's dialect and orthography do not match the Arabic inscriptions of the southern Levant.
His colleague, Marijn van Putten, agrees and goes on to say that the linguistic facts of pre-Islamic Syro-Palestine are irreconcilable with Quranic Arabic.
For van Putten, the loss of the glottal stop (hamza) is another indication that the Quran is not from Syro-Palestine, since such a feature is accosiated with Western Arabia and not Syro-Palestine, since the latter had retained it. One must assume - according to van Putten - an elaborate conspiracy by the Arab grammarians to connect the loss of the hamza with the Hijaz and not Petra if one wants to locate the Quran somewhere in Syro-Palestine.
As for sources, please consult: Robert M. Kerr, “Aramaisms in the Qurʾān and their Significance,” in Christmas in the Koran: Luxenberg, Syriac, and the Near Eastern Judeo-Christian Background of Islam
, ed. Ibn Warraq (USA: Prometheus Books, 2014), pp. 145–234; Guillaume Dye, “Traces of Bilingualism/Multilingualism in the Qurʾānic Arabic,” in Arabic in Context: Celebrating 400 Years of Arabic at Leiden University
, ed. Ahmad Al-Jallad (Leiden: Brill, 2017), pp. 337–371.