Learn Polish: Nie wierzę im, bo ich nie znam. - I do not believe them, because I do not know them.

This is because the two verbs demand different grammar cases. The verb wierzyć (to believe) always forces its object into the Dative case, so "them" becomes im. On the other hand, the verb znać (to know) usually takes the Accusative case, but because it is negated with nie, it must use the Genitive case instead (a rule called the Genitive of negation). The Genitive word for "them" is ich.

If I wanted to emphasize "them" in the first part, would the word im change?

No, the Dative plural pronoun im stays exactly the same whether you emphasize it or not. This is different from the singular Dative pronouns, which have short and long forms for emphasis (like the normal mi versus the emphatic mnie). To emphasize im, you would simply say it with more stress or move it to the front of the sentence: Im nie wierzę! (It's them I don't believe!).

Wait, isn't ich also the Accusative word for "them"? Would it still be ich if the sentence was positive?

It depends on who "them" is! The word ich is indeed the Accusative form, but only for groups that include at least one man (masculine personal). If you said "I know them" about a group of women or objects, you would use the Accusative je (znam je). However, because our sentence is negative (nie znam), we are forced to use the Genitive case. The Genitive form for "them" is ich for absolutely everyone, regardless of gender.