Here are steps of code contribution from the topic "How do I contribute to other's code in GitHub?"
Is it possible to make a pull request without forking a repo?
It's quite sad to see more than 20 repos in my account which were forked for some pull requests. They have no activity then and zero stars / followers. Looks like I'm just copying somebody's code for a black day.
If you don't have access to create branches on that repository, there is no way to create a pull request without forking.