This question is highly non-trivial. The usual strategy is too determine the N'eron-Severi lattice of $X$, determine all effective -2 curves and then determine all possible divisors $F$ consisting of sums of such -2 curves and $F$ is one of the fibers in Kodaira's list. Recently Abhninav Kumar did this programme for Kummer surfaces of jacobians of genus 2 curves.

However if the Picard number of your surface equals 2 then there should not be any elliptic fibration on $X$.

In any case you have first too determine the Picard number of $X$. Upper bounds can be found by considering $X$ modulo several primes $p$, lower bounds can be found by looking for explicit curves on $X$. See Calculations of Pic^0, Pic, NS of surfaces

This question is highly non-trivial. The usual strategy is too determine the N'eron-Severi lattice of $X$, determine all effective -2 curves and then determine all possible divisors $F$ consisting of sums of such -2 curves and $F$ is one of the fibers in Kodaira's list. Recently Abhninav Kumar did this programme for Kummer surfaces of jacobians of genus 2 curves.

However if the Picard number of your surface equals 2 then there should not be any elliptic fibration on $X$.

In any case you have first too determine the Picard number of $X$. Upper bounds can be found by considering $X$ modulo several primes $p$, lower bounds can be found by looking for explicit curves on $X$. See Calculations of Pic^0, Pic, NS of surfaces