@Cranium Clamp, yes. One can show it by the Torelli theorem for K3 surface.

I know that there are infinitely many non-isomorphic exceptional K3 surfaces. I guess that there may be infinitely many non-isomorphic quartic exceptional K3 surfaces but I cannot give a clean explanation. I am wondering if somebody can.

By "non-isomorphic", I mean non-isomorphic as compact complex surfaces. However, if the answer to the original question in the post is not easy, I also would like to know if there are infinitely many quartic exceptional $K3$ surfaces that are not projectively equivalent.

@abx, you're right. It is corrected now.