Sounds to me like you know the answer to your question, you're just looking for someone to validate your choice.
You should never stay in any job where you are unhappy.
You spend too much of your life at a job to be unhappy their too.
Should i quit my au pair job? I only have 2 months to go. I have an exam in a month and i will travel to my home country and come back here for 1 more month. But, i don't want to come back. I hardly have friends here and I'm unhappy.
I think the best solution for you is to quit. You are doing something that you don't enjoy and the exam is approaching so you need to focus on it. Being an au pair is great, and maybe after the exam you can find some other place where you can work that you can enjoy. Good luck and have a great day!