I generated an SSH key pair without a password and added the public key to GitHub.
Connection with
user@dev:/var/www/project# ssh -T git@github.com
H
As usual, create an SSH key and paste the public key to GitHub. Add the private key to ssh-agent. (I assume this is what you have done.)
To check everything is correct, use ssh -T git@github.com
Next, don't forget to modify the remote point as follows:
git remote set-url origin git@github.com:username/your-repository.git