Hello, Elize --
What an interesting piece you have. The obverse appears to be an Indian Head $2.50 quarter eagle, but the reverse, or back, contains the United States Presidential Seal. I don't know if the back of the coin was altered with that design or if the Presidential Seal image is part of the setting itself. What I can tell you for sure is that an unaltered 1914 Indian Head $2.50 quarter eagle gold coin is worth roughly $250 to $300 USD, though the piece of jewelry may be priceless to you and your family.
While I have no idea how this beautiful piece wound up in your family, it's quite possible it entered by way of somebody who had an allure for the 20th-century notion of the so-called "American Dream." Perhaps those hopes and wishes of future prosperity associated with the United States were just one reason that piece of jewelry may have held significance to the person in your family who originally owned it.
All the best,
Josh