"Sing a note" is not a saying.
"sing a note" = to sing correctly hitting every note.
"note" = a single sound at a particular level, usually in music, or a written symbol which represents this sound.
"Strike a note" is a saying.
"strike a note" = to express and communicate a particular opinion or feeling about something.
Example: "I find it really difficult to strike the right note when I'm writing job applications."