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Answer:
A tough situation to be sure, because you can’t compel your friends to raise their child Catholic, even though they had her baptized and made you her godfather.
You mention that her father is an atheist but also your best friend. Encourage him to speak with his wife about allowing his daughter to receive her First Communion, including out of respect for your solemn obligations as a godfather—not to mention hers—since she is apparently the Catholic parent in the family (assuming that your best friend didn’t fall away from the Faith). It sounds like he’s not adamant about raising his daughter an atheist, so hopefully he might consider speaking to his wife, including to help keep peace with her parents, who are presumably practicing Catholics.
If possible, I would also speak to the wife’s parents and any receptive brothers and sisters of hers to see if they might try to counsel their daughter/sister to reconsider. And perhaps also ask her good Catholic women friends to bring the matter up subtly. Is there a priest of whom she is particularly fond? If so, ask his assistance as well.
If these efforts fail, tell your best friend and his wife that you would like to continue to fulfill your role as godfather otherwise, including giving their daughter religious gifts she might like at Christmastime and on her baptismal day and/or birthday. In addition, ask if you might take her to Mass with you and your family periodically, or simply have her/them over to your home for a meal now and then. Just seeing your family pray together before a meal can have an impact. Further, pray in general for your goddaughter and her parents. And perhaps put them on a prayer list(s) without revealing their names.