So to be sure I'm understanding.
Someone with a nosebleed bled directly into your wound? How does that even happen?
There are all kinds of STIs. Do you know what your friend has? My guess is if that they have something that is transmitted by blood (HIV, viral hepatitis), they would be diligent about making sure their blood doesn't get on anyone else. If your friend has chlamydia, gonorrhea, herpes, HPV, syphilis, molluscum or trichomoniasis, you are not at risk at all, as those are not blood borne. They are only transmitted by actual sexual activity.
If they do have HIV or a viral hepatitis (hepatitis B or C), chances are really good that you wouldn't get it this way. (Hepatitis A is spread via fecal matter, not blood, so that's not an issue here.)
First, how old was your wound? If it had started to scab over or close, that's protecting you from getting anything in it and getting infected.
Second, in order for anything blood borne to transmit, there has to be enough blood present, and enough virus in the blood to transmit. If your friend is undergoing treatment, it's quite likely that there wouldn't be enough virus to transmit to you in a single drop of blood. Even if they aren't in treatment, there probably isn't enough virus present to transmit.
HIV dies very quickly once outside it's human host. Hep B and C can live longer, but I'd be very surprised if your friend has either of those. Most people are vaccinated for hep B now - you probably are, depending on your age - and Hep C is usually transmitted by sharing drug use equipment.
So ask your friend nicely if they have an STI if you've actually heard that they do. Be gentle and honest, and tell them that you heard a rumor that they do. (If it's a rumor, I'm going to guess it's not true - most of the time, it isn't.) Tell them that ordinarily, you know it's none of your business (because unless you are having sex with them, it isn't), and that you aren't judging them, but want to clear it up since they had the nosebleed and their blood got in your cut (which I still don't understand how that happened).
If they say they don't have one, believe them, let it go, and never repeat this story. If you hear the rumor, stand up for your friend.