In Gauss-legendre integration we need to find zeros of the legendre function but i can't find a way to write code that enable me to do that? I understand there are list of "xi"s out there by which this function equals to zero but can we write program that find these "xi"s on it own?
In general, finding zeroes of an arbitrary function is not a problem that can be solved by an algorithm. Whether your particular function or class of functions can be solved by an algorithm or not must depend intimately upon the specific mathematical properties of your functions, since your algorithm must rely on those properties specifically. Questions on your functions and what properties it might possess is out of scope for programming and probably computer science as well; I suggest math stack exchange. If you can get a mathematician to explain how to solve the problem by hand, at least in theory, then you are at the point where programmers and computer scientists can start to help you.