Look into the Prometheus jmx_exporter (
https://github.com/prometheus/jmx_exporter). You may still need to change your code to add new MBeans if the metrics you are interested in are not available by default; Spring makes this easy.
The jmx_exporter can be bundled with your application as part of your k8s deployment, or can be loaded in via a sidecar (see, for example,
https://github.com/chuyeow/prometheus-jmx-exporter-kubernetes). In either case, you would need to add an option to the java command line to start the jmx_exporter agent with your application.
It's possible there's a simpler mechanism with Spring boot, but I haven't looked for one (we currently run applications in Apache Karaf). (We also use the Java client, but you mentioned that you do not want to modify your code.)