Assalaamu alaykum. Is the job of quack doctor (illegal doctors, according to the Indian government) permitted in Islam or not? May Allaah reward you.

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah and that Muhammad, sallallaahu ʻalayhi wa sallam, is His slave and Messenger. 

We do not know what you mean exactly by 'quack doctors'. If you mean those who practice medicine without knowledge of medicine, then this is definitely impermissible. The Islamic sharee'ah emphasizes that doctors are held responsible for any medical mistakes and harm they cause because of lack of knowledge or neglect on their part. ʻAmr ibn Shuʻayb narrated on the authority of his father, from his grandfather (the Companion, 'Abdullaah ibn 'Amr ibn al-'Aas) that the Prophet, sallallaahu ʻalayhi wa sallam, said, “Whoever gives medical treatment with no prior knowledge of medicine is responsible (for any harm done).” [Abu Daawood, Ibn Maajah and An-Nasaa'i]

Ibn Al-Qayyim wrote about the doctor ignorant of medicine, “According to the sharee'ah, the ignorant doctor is liable. If he deals with medicine and practices medicine without any previous knowledge thereof, then he is causing harm and death because of ignorance; he is being negligent by recklessly practicing that of which he has no knowledge, and he is cheating the sick person; therefore, he must be held liable according to the majority of the scholars.” [Zaad Al-Maʻaad]

However, if you mean that these doctors are knowledgeable of medicine yet have not received their legal license from the government that stipulates getting a license to practice medicine in order to preserve the public interest and prevent potential abuse and fraud, then they should abide by the government's terms and regulations.

Allaah knows best.