All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallahu ‘alayhi wa sallam, is His slave and Messenger.
There is no religious impediment to keeping and breeding peacocks at home as long as the person takes good care of them, providing them with food and water and the like. The basic principle is that it is lawful to keep and breed animals as long as they are not harmful or ritually impure such as pigs or the animals that are forbidden to keep, such as dogs that are forbidden to keep without a valid reason under Sharee‘ah.
Anas, may Allah be pleased with him, said, "The Prophet, sallallahu ‘alayhi wa sallam, was the best of all people in character. I had a brother who was called Abu ‘Umayr. I think he was newly weaned. When the Messenger of Allah, sallallahu ‘alayhi wa sallam, came to our house, he would say, 'O Abu ‘Umayr! What did the Nughayr (the little nightingale) do?' It was a little nightingale he used to play with." [Al-Bukhaari and Muslim]
Al-Haafith Ibn Hajar said, "It is deduced from this hadeeth that it is permissible to spend money on lawful means of amusement for kids, and it is also permissible to keep a bird in a cage or the like. "
The peacock is a bird. Muslim scholars underlined that it is permissible to sell and buy peacocks for the purpose of enjoying looking at their beautiful colors. An-Nawawi wrote about the animals whose sale is lawful, "There are two categories of ritually pure animals: the animals which are beneficial for humans, like cattle ... and birds, like pigeons, sparrows, and eagles ... and birds kept for the purpose of enjoying their color, like peacocks, or for their voice, like starlings..." [Rawdhat At-Taalibeen]