All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
There is no verse in the Quran that states that cold water is a cure, as your brother claimed. However, it might be that he refers to what is mentioned in the Quran about the Prophet Ayoob (Job) as Allaah ordered him saying (what means): {[So he was told], ‘Strike [the ground] with your foot; this is a [spring for a] cool bath and drink.”}[Quran 38:42] Hence, he stroke the ground with his foot and a spring of cool fresh water sprung, and he drank from it and had a bath. Thereupon, he was cured from every inward and outward disease, as mentioned by the interpreters of the Quran. This is a mercy from Allaah to his slave Ayoob, as Allaah Says (what means): {...as mercy from Us}[Quran 38:43]
However, this does not mean that every kind of cool water is a cure. On the contrary, a healthy person may drink cool water and get sick.