Call your contact at the company, either your boss or the HR. Tell them you will be on sick leave. It would be good to have at least a somewhat specific date. For example "I will have to undergo a procedure on January 4th and the doctor told me that recovery can take up to four weeks". You don't need to get specific about what that procedure is.
Please note that you don't ask for sick leave in Germany. This is not a favor they do you or something they would need to approve. You are entitled to it. You will be on sick leave. Period. Maybe be polite and say you are sorry this happened. But that's it.
Make sure you have an "Arbeitsunfähigkeitsbescheinigung" (commonly abbreviated "AU" or called "gelber Schein" ("yellow slip", because the paper used to be yellow, don't be surprised, it is pink-ish now) for any day you miss. You get this certificate from your doctor. It is not uncommon to get that in stretches, lets say you get one for the procedure and the week after, and then another for a week and then another for a week and then maybe one for two days and then you go back to work if the doctor deems you fit to do so.
Your employer does not pay you if you get sick in the first four weeks of your employment. Instead, the health insurance will pay you.
So don't propose to move your start date. It will do nothing for your employer, who will not pay you either way, and only cheat you out of your benefits with the health insurance. Because if you are unemployed of your own free will, you will get nothing.
This should be just a little administrative issue. Germany has rules for it. If anybody at that company gives you anything else than a "best wishes/hope you get well soon/don't worry about it" reply, this would be highly unprofessional and I would reconsider working there altogether.