This is better late than never.
You can trigger from the audit log.
You want to create something like the following:
gcloud eventarc triggers update <trigger-name> \
--destination-run-service="<function name>" \
--destination-run-region="<region of function>" \
--location="<region of trigger and bucket>" \
--event-filters="methodName=storage.objects.create" \
--event-filters="resourceLocation=<region same as location>"\
--event-filters="type=type=google.cloud.audit.log.v1.written"
--event-filters-path-pattern="projects/_/buckets/<your bucket name>/objects/<your filter pattern>"
Further reading available here:
https://cloud.google.com/eventarc/docs/path-patterns
Not all services offer the path-pattern filtering, so you need to run something like this to get the details:
gcloud eventarc providers describe cloudaudit.googleapis.com
I hope that helps, 6 years later :D
Feel free to update to relate to firebase more specifically or mark as answered.