I am currently experimenting with the Google App Engine flexible environment, especially the feature allowing you to build custom runtimes by providing a Do
You can follow following steps to create a container with docker-compose file in Google App Engine. Follow link
You can build your custom image using docker-compose file
docker-compose build
Create a tag for local build
docker tag [SOURCE_IMAGE] [HOSTNAME]/[PROJECT-ID]/[IMAGE]
Push image to google registry
docker push [HOSTNAME]/[PROJECT-ID]/[IMAGE]
deploy Container
gcloud app deploy --image-url=[HOSTNAME]/[PROJECT-ID]/[IMAGE]
please add auth for docker commands to run.