Is it true that it was reported from the Prophet that a mother is forgiven all her sins by giving birth?

There is no Hadith (definition) with this precise wording. However, a woman's pregnancy, childbirth, patience, and seeking reward from Allah Almighty are all reasons for the expiation of her sins and the removal of her transgressions. If an ordinary illness expiates sins, then pregnancy, with its duration and intensity, is even more deserving of that. Illness is a reduction of sins and an expiation for them, and patience in enduring it is an act of obedience and merits reward. The Prophet, peace and blessings be upon him, said: "No fatigue, nor disease, nor sorrow, nor sadness, nor hurt, nor distress befalls a Muslim, even if it were the prick of a thorn, but Allah expiates some of his sins by it." And he, peace and blessings be upon him, said: "No Muslim is afflicted with harm, be it illness or otherwise, but Allah causes his sins to fall away as a tree sheds its leaves."