When creating a service by Google Cloud Run, a trigger run is not done
Solution 1:
I fail to understand what is your issue, mainly due to language.
Cloud Build documentation has a section on creation and management of triggers using GitHub, which I assume can fit your case.
Cloud Build is a service that executes your builds on Google Cloud Platform infrastructure. Cloud Build can import source code from Cloud Storage, Cloud Source Repositories, GitHub, or Bitbucket, execute a build to your specifications, and produce artifacts such as Docker containers or Java archives, so obviously there is no run button there.
To see how to create a Cloud Run service using a Cloud Build image you can follow this quickstart.
To attach the previously created Cloud Build trigger to a Cloud Run service, you can see this documentation.