Did I accidentally remove a piece of cartilage from my nose 3 months after rhinoplasty?

The answer is that it may or may not be; cartilage grafts don't typically extrude 3 months postoperatively.  The most useful advice that can be given to you is to return to your Surgeon.

I doubt that at three months after rhinoplasty you removed a piece of cartilage. If concerned of course you should discuss this with your surgeon.

At 3 months post op, the only way to answer this question is with an exam.  Follow up with your Plastic Surgeon, as he is  will know what was done in your surgery, such as cartilage grafts, etc.