What is the meaning of the noble verse: "A Qur’an recited in the hours of the night and the ends of the day," and what is the time during which the Qur’an is recited in the verse?

What is mentioned in the question is not a Quranic verse. Perhaps the questioner is referring to the Almighty's saying: "They are not [all] the same; among the People of the Scripture is a community standing [for the truth], reciting the verses of Allah during periods of the night and prostrating." (Al-Imran: 113), which means that the believers among the People of the Scripture are not the same as the criminals among them; rather, among them are the believers and those who obey Allah. Or perhaps she means the Almighty's saying: "And from [periods] of the night, glorify [Him] and at the ends of the day that you may be satisfied." (Taha: 130), which means the command to be patient and glorify [Allah] during virtuous times. The meaning of both verses is general for anyone who is characterized by what is mentioned therein.