This previous StackOverflow question addresses how to move your repository from another service over to GitHub, the first answer there addresses how to do it via command line, and the second and third are more user friendly ways, which unfortunately will not work for you if your GitLab instance is on your local server (which seems to be your case).
You can however 'import' your repository from the command line to GitHub as explained by GitHub docs, this is the suggested way as GitHub offers this as an alternative to using their GitHub Importer tool (which is highlighted in that previous SO question)
A run down of steps as taken from the documentation:
- Create a new repository you want to push to in GitHub.
Make a local bare clone from your GitLab server:
git clone --bare https://githost.org/extuser/repo.git
A bare clone is an exact duplicate, without a working directory for editing files, so it's a clean export.
Change into that directory and then push it with the
--mirror flag. The mirror flag ensures that references (branches/tags) are copied to GitHub.
git push --mirror https://github.com/ghuser/repo.git
Finally remove the local repository you made.
rm -rf repo.git