I wanted to ask how authentic is the sanad of ijazah in qur'an?


I wanted to ask how authentic is the sanad of ijazah in qur'an. How did it come about? And do we know its true? Also what if someone does not have ijazah in qur'an but knows how to read qur'an well?  Can he still teach people?

Wa alaikum assalaam wa rahmatullahi wa barakatuh. 
The authenticity of any ijaazah depends on the sanad, who is in it, was the teacher alive at the same time as the student who claims to have read to him/her, were they at the same place at the same time, were their witnesses, etc.  This authentication has been researched by the scholars of the Qu'ran and there are books that prove different sanad or show a specific sanad to be in question.   If it is in the present day, it is easy to check if the ijaazah is a true ijaazah or not, by asking the sheikh giving the ijaazah if he/she gave this ijaazah.  Always check which type of ijaazah the sheikh/sheikhah gave to the student (present day) to make sure the ijaazah is in recitation and not something else. 

The Prophet, peace and blessings of Allah upon him approved the reading of different sahabah after listening to them recite the complete Qur'an by memory, these sahabah listened to and approved the reading of those after them, and so on.  There are well documented authentic chains from the Prophet to the sahabah he approved, peace and blessings of Allah upon him down through the chain until we reach the present day shuyookh of the Qur'an. 

One does not need to have an ijaazah to teach the Qur'an, but of course it is a higher level if one obtains one.  It is recommended though, that a  teacher gives a recommendation to a student of the Qur'an to teach, before they teach.  This gives both the teacher and student peace of mind as to their qualifications.