Is there anything (incumbent) upon a woman who gave birth in Ramadan, broke her fast on the Day of Arafah in Dhul-Hijjah, and completed her Dhul-Hijjah fasts in Muharram, due to her fasting the month of Dhul-Hijjah after hearing that it is not permissible for her to fast it?

There is no harm in fasting the month of Dhul-Hijjah with the intention of making up missed fasts or as a voluntary fast, as it is one of the Sacred Months in which fasting is recommended. It is not permissible to fast the Day of Sacrifice (Yawm al-Nahr), even as a makeup fast, nor the Days of Tashreeq except for the pilgrim performing Tamattu' Hajj who could not find a sacrificial animal. According to this view, fasting these days does not count as making up missed fasts.