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)
我们没有共同语言。这是地道的中文翻译方式，不是直译的。你也可以翻译成“我们没有共同的兴趣”。只是日常口语一般不会说的这么正式。
我们没有共同爱好.
A local expression is : 我们玩不到一块儿去. (note that 儿, you should use 儿话音 to speak this sentence, or the feel will fail).
or a wild expression: 我跟你尿不到一个壶里. means you are two types, and cannot get along with each other. The opposite is 我跟他穿一条裤子. means you are very close. Don't say 我跟 你 穿一条裤子 to someone, that means a totally different thing, except you really mean that.
First of all, you saying this to someone's face? What is the context of this sentence?
In my experience people in China almost never say such things to each other, cus it's just not very polite. Cultural difference. Therefore any translations would probably come out wrong for a Chinese.
I'd cut the sentence in 2, like this:
我们好像不能沟通。兴趣爱好完全不一样
沟通 is just the word you need in this kind of situation. it's about the ability to find a common ground (or not), between persons but also between cultures (西方人和中国人思想不一样！). Mostly translated as 'communicate', i'd rather put it as 'relate to'.
好像 ('it seems') softens up the message.
完全 underscores the difference
