检查是否语法错误？I firmly know that one day you will open your heart for me.

除了楼上指的介词搭配问题外，语法上应该没有错误。但用firmly搭配know 在英语表达上的确显得很不地道，是中国化的英语吧。：）你可能想表达的是“深知”或“坚信”之意吧。
可以改为: I firmly believe that you will open your heart to me one day.
或：I am sure that you will open your heart to me one day.

我觉得I know firmly that one day you will open your heart for me.会更好些吧。当然你原来的句子也没什么问题

I know firmly that you will open your heart to me one day.

open （one's） heart to sb