Is there a general method to solve problems based on finding number of integral solutions, or do should we try out whatever works each time? For eg. What is the number of positive integral solutions for x^2 + 3y^2 = z^2

Hi 
There is no general method to solve such questions. Each question requires some rearrangement to make it easy to solve.
In the given questio, we can write it as
x^2 + y^2 + 2y^2 = z^2
We can see that if x = y, then we have
(x+y)^2 = z^2
Thus, if x+y = z , we will have a solution.
Hence we can have infinite solutions for the given problem.