Can I ask my friend not to bring up anime in public?
Why do you not want your friend to bring up anime in public?
It's because I find anime stupid and the reason to hate it
If you dislike anime that much then it should be expected that your friend respects that. It can be embarrassing to force yourself to listen to something you don't like.
I'm going to ask my friend not to bring up anime in public then.
I think that is fine, as long as you do so in a respectful manner, your friend should respect that.