I know I can say ときどき for sometimes and よく for often, but would よくじゃない be a correct way to say “not often”? If not, how would one say “not often”?
While 「よくじゃない」("Not often."), all by itself, might actually be uttered by some in very informal conversations, it would also be considered to be fairly sloppy speech by others. It sounds kind of curt all by itself.
I would recommend that you add a little word or two to it to avoid that and say instead:
You would sound like a much better conversationalist if you said one of the above instead.
“not often” is usually translated as "あまりしない". I don't think "よくじゃない" is common phrase. However, when you are asked "よく旅行する?", it wouldn't be unnatural that you answer like "よくじゃないけど、たまに。".