Can an Indian Sunni Muslim married in India obtain divorce in Qatar? If yes, what are procedures and what are the implications if the couple has three children below 8 years old?

Praise be to Allah, the Lord of the World; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

We advise you to refer to the Islamic court in Doha; they will provide you with the information you need since the court is a liable authority in this matter.
However, the matter of nursing and guardianship of children after divorce is made clear in Fatwa: 83052.
Allah knows best.