Well, by intuition, of course there is doesn't exist any nonzero integers, but how would you prove that? I was thinking of doing the GCD of $a$ and $b$ is $1$, but that leads me to nowhere.
It doesn't imply that they are prime. It means that a/b is then positive or negative root 3. Let's just take it to be positive. You now have a ratio of assumed integers giving you an irrational number, root 3. Do you think this makes sense? What conclusions can you make from this?