As for the abominableness of recitation of the Quran by a menstruating woman, it does not make any difference whether she recites the Quran in the month of Ramadhan or in any other month; reciting the Quran remains makrooh for her.
Late Imam Khomeini (may Allah bless him) says in this regard: “It is abominable for a menstruating woman to recite the Quran even though less than its seven verses, to carry the Quran even with its cover, or to touch the margin of the Quran and the space between its lines.”
It is necessary to note that abominableness in the acts of worship means ‘gaining less reward’. That is to say, there is reward in the recitation of the Quran by a menstruating woman but when it is compared to the time when she is not in her menses, it has less reward. Therefore, if a menstruating woman recites the Quran in the month of Ramadhan, she gains the reward but she will gain less reward in comparison to the time when she recites in the month of Ramadan in the state of cleanness and ritual purity.
 - Tawzih al-Masail (with annotation by Imam Khomeini), vol.1, p. 269, issue No. 477.