We have had a thirteen year old girl of no relation to either of us, in our home for two years. Neither one of her parents has called or tried to contact this girl in the last thirty days. Would we be able to file for abandonment under Utah state law?

Not really, and this can turn bad on you very fast ?

1. how did you come to have the girl in your home ?
2. was her parents aware she was in your home? Have
3. you have no legal right to put her in school, get her medical care or more. So it is very likey you have broken several laws having her for two years.
4. One can not just keep someone else's child for two years

Yes you need to report this to social services, but it is very likely the child will be taken into custody.

What you need is to find the parents and get legal guardianship signed ove