How do you say "we don't have any common interests"? I was trying to say this to someone and it came out "我跟你们没兴趣", which I can tell isn't right.

To clarify, common interests here refer to personal interests or interests in activities, rather than shared goals in the context of business partnership or political alliances.
"interest" has multiple related meanings and that's why Ghost's answer struggles. OP does provide some context but not enough.

Under the context of personal interest, common interest will be 共同的兴趣爱好 something we both like.

Under the context eg "The common interest between US and China" that will indeed be 共同的利益.

(In a constructed case, "my bank doesn't have common interests with his bank" this can even mean 一样的利息)