They are not wrong but they're not the most natural either. 여행하다 is a verb used in more formal contexts. And we say 한국을 여행하다 with 을/를 for this verb. In day-to-day speech, we usually say (-에/으로) 여행(을) 가다. So you can say it like these (personally, I think the two are equally clear and natural). - 한국(에) 여행 가는 것을 좋아해요. - 한국(에) 여행 가기를 좋아해요.

The first one is a right sentence in terms of grammar. And the second one simply doesn't make sense. But what the important thing is we know what you meant by both of them.

The grammar is correct in both cases, but the with 여행 you must use 가다 instead of 하다.