Is she unhappy now, or are you worried that she will be unhappy?
It always seems to me that when it comes to friends, the biggest difference is between having 0 friends and having 1, especially for introverted gifted kids, who are by their nature less likely to fit in well with age-peers, are less likely to find intellectual peers, and are less driven to seek out friendships at all.
Also, gifted kids often have activity-specific friendships that don't go beyond the activity: a chess friend, a softball friend, a friend to play dolls with, etc. This also doesn't necessarily indicate a child who is unhappy or lonely.
So does she have a friend of any sort in her life? Or is she alone besides her family? This friend doesn't necessarily have to be a school friend, though having a friend at school might reduce the chances that she'll be bullied.
There's nothing inherently wrong with being a shy introvert, and the child you describe doesn't sound unhappy on the face of it (though you do say she's "starting to notice" that she's somehow different...). The parents' accepting that an introverted personality is okay can go a long way to helping a child to be willing to grow socially.
My recommendation: find group activities that are appropriate to her interests and intellect. Maybe a kids' book club? Or a gifted support group? She may find someone there with whom she can connect. But don't push. Ask questions about the people she's around, but don't lead her to feel there's something wrong with her if she's not really interested in friendships with the kids she knows. She may just be shy and selective. There's nothing inherently wrong with either of those things!
