Is it obligatory to seek forgiveness from one's mother and confess to her the act of sending her pictures to people, or should one conceal the matter and leave it to God, especially since the mother is ill, after the son has repented and regretted his action?

What you did by spying on your mother and attempting to expose her is a serious matter and a great act of disobedience (ʿuqūq). A mother's wrongdoing does not justify her child mistreating her. You have done well by repenting of this crime. Among the conditions for repentance from disobedience is to seek your mother's forgiveness and apologize to her. However, if doing so would cause greater harm—especially given her illness—then it is sufficient for you to frequently supplicate for her well-being and strive to show her utmost piety and kindness. You must remove the traces of this sin, such as the pictures you sent; anything beyond your capability is excused. If you are sincere in your repentance, Allah will accept it, for He is the Forgiving, the Merciful. He says: "And indeed, I am the Forgiving to him who repents, believes, and does righteousness, and then is guided." Repentance, uprightness, frequent remembrance of Allah, reciting the Quran, and supplication are among the greatest aids in overcoming psychological pressures.