My mother has heard from someone that one should not read Quran and supplicate during and around 12 noon and 12 midnight times of the day. Please clarify on this matter.

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 is His slave and Messenger.

It is permissible to supplicate and recite the Quran at any time of the night or the day and what your mother heard is not correct. Rather, supplicating and reciting the Quran are permitted at all times unless there is evidence which prohibits it. We are not aware of any Hadeeth which indicates that it is forbidden to supplicate and recite the Quran at the times which you mentioned in the question.

Nonetheless, there is evidence that prohibits performing supererogatory prayers at five different times as mentioned in two Ahaadeeth. The first Hadeeth is narrated by ‘Uqbah Ibn ‘Aamir Al-Juhani who said: “There are three times at which Allaah’s Messenger forbade us to pray or to bury our dead: when the sun begins to rise until it is fully up [about 15 minutes after sunrise], when the sun is at its zenith [at midday] until it passes the meridian, and when the sun draws near to setting until it sets.” [Muslim]

The second Hadeeth is narrated on the authority of Abu Sa’eed [Al-Khudri] that the Prophet said: “There is no prayer after performing 'Asr prayer until the sun sets, and there is no prayer after performing Fajr prayer until the sun (sufficiently) rises." [Al-Bukhari and Muslim]

Allaah Knows best.