It seems like you have to interact with github.com to initiate a pull request. Is this so?
A man search like...
man git | grep pull | grep request
gives
git request-pull []
But, despite the name, it's not what you want. According to the docs:
Generate a request asking your upstream project to pull changes into their tree. The request, printed to the standard output, begins with the branch description, summarizes the changes and indicates from where they can be pulled.
@HolgerJust mentioned the github gem that does what you want:
sudo gem install gh
gh pull-request [user] [branch]
Others have mentioned the official hub package by github:
sudo apt-get install hub
or
brew install hub
then
hub pull-request [-focp] [-b ] [-h ]