Is it permissible to announce the Tahajjud prayer by reciting the Qur’an via the mosque’s microphone in Ramadan or otherwise?

It is not prescribed to alert people for Qiyam al-Layl (night prayer) via mosque loudspeakers, whether by reciting the Quran or otherwise, neither in Ramadan nor at other times. If it were prescribed, the Sharia would have legislated it. The Prophet, peace and blessings be upon him, said: "Whoever introduces into this affair of ours that which does not belong to it, it will be rejected." Therefore, calling for Qiyam al-Layl in this manner is an innovation in religion and a rejected act.