Wa `alaykum As-Salamu wa Rahmatullahi wa Barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
Dear sister in Islam, thanks a lot for your question, which reflects your care to have a clear view of the teachings of Islam. Allah commands Muslims to refer to people of knowledge to become well acquainted with the teachings of Islam in all aspects of life.
First of all, we would like to make it clear that reading the Glorious Qur’an entails great merits. The Prophet (peace and blessings be upon him) said: “Whoever reads one letter of the Book of Allah will have one hasanah (reward) for doing so, and every hasanah will be multiplied by 10. I do not say that ‘alif, lam, mim’ is one letter, but ‘alif’ is a letter, ‘lam’ is a letter and ‘mim’ is a letter.’” (At-Tirmidhi)
Responding to your question, Sheikh Ahmad Kutty, a senior lecturer and Islamic scholar at the Islamic Institute of Toronto, Ontario, Canada, states:
Recitation of the Qur’an is permitted throughout each and every day, even during the times when optional prayers are prohibited.
The only consideration to be made is regarding the prostration of recitation or Sujud At-Tilawah as some scholars (Hanafi persuasion, etc.) prohibit prostration during the times that prayer is prohibited. The majority of scholars do not find anything wrong in reciting Qur’an at anytime, as the hadith texts relating to prohibited times of worship only refer to Prayer. If reciting the Qur’an had been intended, it would have been clearly stated.”
Almighty Allah knows best.