The Czech sentence translates literally as "that he will" rather than as "that he would." However, formal English traditionally requires the past tense "would" instead of "will" in sentences of reported speech that begin with a verb in the past tense (i.e., "he said" rather than "he says.").
So the answer to your question is that the Czech for both those English sentences is the same, whereas only the "would" version is technically correct English. But nowadays many people would say "will" just as readily as "would" in such sentences.