It's not possible. Suppose I programmed a computer to succesively test every even number, and see if it's the sum of two primes or not, and started the program.
If the program ever stopped, that would mean it had found a counterexample, which would prove the conjecture to be false. BUT, if the conjecture was true (though unprovable) then this could never happen.
So, that means that the program will never find a counterexample... but that proves the conjecture to be true, so it wasn't unprovable after all!
The final conclusion is that the conjecture may be true or false, but we cannot prove that it cannot be proved.