I agree with the previous comment, and I would like to add a few things as well!
There is nothing wrong with being a quiet, introverted person, so personally I believe it is worth asking yourself why you wish to change this about yourself? You said it yourself -- your patients are appreciative of the care you provide, and you are able to adequately provide care and advocate when necessary! What about your practice makes you feel as though you need to change something about yourself that is so innate, such as your introversion? You say it is confidence you need; why do you feel are you not confident? Are you simply not confident enough in yourself to carry on conversations and make friends, or are there other areas you wish to be more confident in? For instance are there things as a nurse you don't know the answers to, or are you struggling in certain areas, like problem solving, therapeutic conversation (obviously if you are shy and not very talkative, this can be tricky!)? Do you feel anxious when having conversations with or making phone calls to physicians? Or are you worried that you are not well-liked by your peers, and do you wish to be part of the group that chit-chats and participates in work-outings?
I ask these questions because as someone who is quite introverted myself, I struggled quite a bit with feelings of anxiousness and a lack of confidence when I first started as a nurse, as well. I was an anxious wreck any time I had to pick up a phone, or walk into a room with more than one family member, or have a crucial conversation with a coworker. It took me well over a year to really make friends with my coworkers, too., though this was not as difficult for me as the other things I mentioned, like phoning physicians or handling a disgruntled family. My struggle with feelings of anxiety has been ongoing, though it has improved dramatically. Yes, that anxiety has affected my confidence at times, but as the months and years go by, the mere exposure to these situations and seeing that I can (as you mentioned of yourself) adequately navigate through sticky situations to achieve my desired outcome, gives me more confidence. And who's to say how much confidence is enough, or not enough, or too much? That is subjective.
My advice to you is to, like I said, examine exactly what it is you wish to achieve. Keep in mind there are a variety of nursing styles, and every nurse has their strengths and weaknesses, and it is this diversity that makes a team great! Some nurses are experts at critical thinking in emergency situations, some are the best therapeutic conversationalists known to man, some can multi-task like there's no tomorrow. Some people can do all of this, but are still not chatty with their peers. Some nurses have awesome relationships with their coworkers, but their patients greatly dislike them. You see what I mean? Figure out what your strengths are and build on them! The more you do this, the better you will become, and the confidence you build as a result will spew over into those other areas where you feel less confident. Start small.
As for your work relations, ask people how their day/evening/shift/patient/dog/family etc have been doing, remember details about them. Compliment people, but only when you really mean it! (i.e. don't be a suck up.) Even if you are a little awkward, they will remember that you are attentive and caring and genuine, and may be more apt to include you in conversation. Recruit your coworkers to help you accomplish tasks. "Hey, Lydia. I know you are excellent at smoothing things over with really upset family members. I have this patient whose daughter hates my guts and thinks I can't pack wounds to save my life, would you mind helping me with this dressing change? I need some back up in there." I'm mostly kidding, but you get the idea. Let your coworkers know when you DO value their strengths; they will appreciate it. The little things add up over time. If they still exclude you, well, maybe you don't want to be friends with them anyway. Let work be work and friends be friends, and don't mix the two.
Anyway, I hope I was able to help at least a little bit. Hang in there! You're doing better than you think :)