Although the book of Esther, unique among all the books of the Bible, never mentions the name of God, the story is a distinctly Jewish one. It leaves no doubt in the minds of its readers that Esther was a devout Jew, nor does it leave room to wonder if Esther believed in God.
Let's just look at some of the elements that make up the Jewish thread that runs through this story.
First, we are told that Esther was raised by her uncle, Mordecai, and that she always did everything Mordecai told her to do. It seems natural to us that a relative might be a great person to leave our children to in the event of our untimely death, but this idea was woven deeply into the Jewish covenant. It was the idea of the kinsman-redeemer - if a man died, his closest relative inherited the responsibility of maintaining his family line. This is usually seen when a man leaves behind a wife with no children, but it is also true where a man leaves behind children with no parents (widows and orphans...sound familiar?).
This is also why, by the way, we see Mordecai and Esther take individual paths in this book. Mordecai does not depend upon Esther to create for him a position in the kingdom because whatever Esther accomplishes is for her father's name, not her uncle's. Thus, Mordecai makes his own position in the kingdom. And Esther does not depend upon the king's favor toward Mordecai to give her any hearing before him; she has to make her own way. If you've ever read this book and wondered why they are whispering in secret at the window rather than depending on each other to get further in their exploits, this is why.
We also see that Esther, upon recognizing that there is something she must do for the sake of her people, enters a three-day fast and asks the Jews of the kingdom to join her. Fasting was a distinctly Jewish exercise, and it would have been extremely common for Jews to fast on an occasion such as this.
Not only that, but we see here an element of "the people." The Jews were a community, and when Esther hears that her community is in trouble, she calls upon her community to stand with her as she prepares to stand for them. The Jews do essentially nothing at all on their own; the entire locus of their identity is communal. So if Esther is a faithful Jew, we should expect this from her. And, indeed, she does not let us down.
Mordecai is also a Jew, which we see first in his willingness to act as kinsman-redeemer in the first place. But then, we get a glimpse into Mordecai's life of faith when we see that he refuses to bow down to, honor, or pray to the statue that the pagans have set up in the courtyard. Any other faith of the time was polytheist, which meant that the other faiths accepted a multitude of gods simultaneously; only the Jewish faith was monotheistic, asserting there was only one God. And it was clear in the Jewish faith that one should not ever even entertain the idea that bowing down, honoring, or praying to any other idea was not to be tolerated. It's clear that Mordecai is a Jew.
A kinsman-redeemer who refuses to bow down to idols and a queen who finds her identity in the collective of her people, with whom she fasts - yes, this story has a Jewish thread running right through it. These elements are at the heart of the Jewish faith, and they are in the hearts of the main characters.
That's why Esther's story doesn't require a mention of God by name; He is woven full through in even the finest detail.