It turns out I needed to first install Java. Here is the final snippet that worked:

- name: gcr.io/cloud-builders/npm
  entrypoint: bash
  dir: edcloud/cloud-function/${_PROJECT_NAME}
  args:
    - -c
    - |
      apt-get update && apt-get install -y openjdk-11-jdk
      npm install -g firebase-tools
      npm install
      firebase emulators:exec --project $PROJECT_ID 'npm test'