Based on what you've shared, I have 2 theories about what might be wrong.

1. (Most likely) Since you didn't provide a full command output from inside container (i.e. curl vs curl ... | grep ...) I can assume that the grep version inside conatiner is working different than expected. This is usually happens with more complex commands (e.g. when using -E), but it worth checking a full piped pair.

2. (Less likely) Weird idea, but maybe YAML itself is not resolved correctly? Try to make it as simple as possible to 2x check:

   startupProbe:
     exec:
       command: ["sh", "-c", "curl -s -f http://localhost:8080/v1/health | grep -q -e '\"status\":\"healthy\"'"]
   

If this doesn't work, try to make it verbose and check the Pod logs:

startupProbe:
  exec:
    command:
      - echo "PROBE DEBUG"
      - curl -v http://localhost:8080/v1/health
      - sh
      - -c
      - >
          curl http://localhost:8080/v1/health |
          grep -e '\"status\":\"healthy\"'
      - echo "$?"