"Of my own accord" means of my own will, without coercion or force. It is the correct form of the idiom. "On my one accord" is technically incorrect, but still used a lot.

I believe both idioms are correct and have the same meaning: Spontaneous or voluntary desire to take a certain action. Example: The children returned on their own accord. He confessed of his own accord.

“of my own accord” is a idiom ，means "by yourself"(nobody helped you ,nobody forced you