Praise be to Allah.
Celebrating one's birthday is an innovation that has crept in to Muslim countries from non-Muslim countries, hence celebrating it is an imitation of those disbelievers. For more information, please see the answer to question no. 1027.
Congratulating someone else on the occasion of his or her birthday is a kind of participating and cooperating in this innovation, and imitating the disbelievers.
Allah, may He be exalted, says (interpretation of the meaning): “Help you one another in Al‑Birr and At‑Taqwa (virtue, righteousness and piety); but do not help one another in sin and transgression. And fear Allah. Verily, Allah is Severe in punishment” [al-Maa’idah 5:2].
Although obeying one's parents is one of the most emphasized duties, it should not be in anything that involves disobedience towards Allah, may He be exalted.
Allah, may He be exalted, says (interpretation of the meaning): “But if they (both) strive with you to make you join in worship with Me others of which you have no knowledge, then obey them not, but behave with them in the world kindly, and follow the path of him who turns to Me in repentance and in obedience. Then to Me will be your return, and I shall tell you what you used to do” [Luqmaan 31:15].
Shaykh ‘Abd ar-Rahmaan as-Sa‘di (may Allah have mercy on him) said:
“But if they (both) strive with you” means, if your parents try hard
“to make you join in worship with Me others of which you have no knowledge, then obey them not” and do not think that this comes under the heading of treating them kindly, because the rights of Allah take precedence over the rights of all others, and “there is no obedience to any created being if it involves disobedience towards the Creator.”
He did not say “If they (both) strive to make you join in worship with Me others of which you have no knowledge, then turn against them”; rather He said “then obey them not” that is, by joining others with Allah in worship. As for honouring them, you should carry on doing that. Hence Allah says “but behave with them in the world kindly” that is, keep them company and treat them kindly. But with regard to following them in their disbelief and sin, do not follow them.
End quote from Tayseer al-Kareem al-Mannaan fi Tafseer Kalaam al-Mannaan (p. 648)
In accordance with this Qur’anic guidance, you should explain to your mother very politely the reason why you do not congratulate her on her birthday, and strive hard to honour her and treat her kindly in anything that does not involve sin. One of the greatest ways in which you can honour your mother is to try your hardest to call her to adhere to the laws of Allah, may He be exalted, and to pray a great deal that she be guided; then soften her heart towards you by giving her gifts, treating her kindly, honouring her and getting in touch from time to time, whenever it is possible for you to do that.
For more information, please see fatwa no. 26804.
And Allah knows best.