I'm not going to tell my cousin happy birthday
Why would you go without telling your cousin happy birthday?
I am really annoyed by her whining. It is starting to make me feel pretty angry.
If you aren't a fan of someone you don't need to interact with them just because they are family. Even if someone is related to you that doesn't mean you need to give them the time of day.
Well, I don't really want to talk to my annoying relatives.
Nobody is forcing you to spend time with this family member. That being said, you can still tell her happy birthday and be the bigger person here. She might be appreciative and may even have a change in attitude towards you.