The size and stability of the company is likely to influence attitudes. A large company that plans it's workforce ahead won't miss you for a month, and won't miss the cost of your salary.
A small company that's desperately needed a new member of staff for some time, would probably be in uproar.
The importance of the surgery, and the long-term implications for your health at work, will also probably influence attitudes. Employers are probably more sympathetic to a new employee going off for knee surgery, than they are to either a cosmetic hair transplant, or cancer.
There's one obvious question, which is why you changed jobs so near to such an event? Or was it compulsory due to circumstances?
It's unclear overall how urgent or important you consider the surgery to be.
There may be local laws that protect you from being fired in these circumstances. I can't imagine it affecting career progression, from the year mark onwards. A month's leave for legitimate surgery would likely have been forgotten by then, even if it was inconvenient at the time.
The best thing to do is to ensure you have a conversation as soon as possible about booking the time off. 3 months' notice is generally ample, and it's not soon as to disrupt usual induction arrangements in a new job, or so soon that it would have made sense to delay the start date until afterwards.
If you've genuinely mentioned the need for time off earlier in the recruitment process, then it's reasonable to approach your manager (or HR) in the tone of merely formalising in the calendar what you've already notified.
It's reasonable to take this approach even if you know the information wasn't, in fact, relayed from the person you informed. An absence of significant concern and internal fuss about such medical leave may be a sign of a responsible employer.
However it may be a harder conversation to have if you didn't mention it earlier at an appropriate time (such as on a form asking whether you had any fixtures in your own calendar like holidays). In this situation, it might be better to open the conversation in the tone of asking permission to take time off, and justify it on the basis that you've tried to rearrange the surgery for the future but have found that it would be unreasonably delayed. They may have no real grounds to refuse, but it avoids presenting the need for imminent leave as an accomplished fact, and it avoids suggesting that it was already a fixture since it can in theory be rearranged.