How many syllables are in the infection?

If you are asking how many syllables are in the term "the infection", then the answer is four. "The" has one syllable, and "infection" has three: In-FEC-shun.

If you want the exact answer, we'd need to know which infection you are asking about. And even then, there are no syllables in any infection: just bacteria and blood and gunky stuff like that. "The" is a definite article, which means you are asking about a single specific infection.