Proving gods existence

reductio ad absurdum . Just tell her that Bilbo Baggins exists because it is written in the Lord of the Rings.

Same applies for Spiderman and Harry Potter.

Hmm... This can be circular reasoning, but you know they will play these tricks like, " how do you know that indonesia exists? And then you will obviously tell that people saw that exists and people said so, and hence god in the bible is true, because Jesus has experienced god, but you can counter that by saying god of bible is false because other religions tells us that they are true.

By explaining to her that claiming God exists, or claiming that God doesn't exist is an unfalsifiability fallacy. The only logical and reasonable position to take on any unfalsifiable claim is to be an agnostic. This is because you cant prove or disprove an unfalsifiable claim, therefore its unknowable and therefore you must default to the agnostic position. But first approach her with, "can you prove God doesn't exist" and see what she says. She will likely say no, then you can follow it up with "well then you also cant prove he exists" then explain the fallacy. Unfalsifiablity fallacies often lead into circular reasoning  fallacies to try and justify them. 

Methinks you are being sucked into a discussion that can have no conclusion other than to cause distress between you and the other person. If you want to engage her, ask questions (a la Boghassian) about what drives her to her conclusions. Do not confront her.

If you want to sharpen your own debate skills or more clearly think about the subject in your own mind, or have some fun, go ahead, but start at the beginning. 

Too often people enter a discussion about God before anyone in the conversation knows what the subject is. The debate begins too high in the hierarchical logical structure, which ensures false premises and the resulting confusion among the participants.

Of the almost five thousand known named gods to which humans have groveled before, feared, spoken and prayed, one must ask which god is under discussion. Usually, this ends up being about the Abrahamic god, Jehovah. Abrahamism is a group of allegedly monotheistic religions claiming to worship Abraham's God (Jehovah), including Judaism, Samaritanism, Christianity, and Islam.

As a side note, what confuses me is the Christians claim four gods (not counting the evil god they call Satan) to be one, thereby maintaining their monotheism. The four are: The Holy Impregnator, Mary the Impregnated, Jesus the Progeny, and the Big Guy, who got his kicks watching the impregnation process, Jehovah. 

But, I stray. Now back to the subject. To have a rational conversation about whether a god, any god, exists, one must first have an intelligible definition of that god. Unfortunately, nobody ever gave me a rationally intelligible definition of a god. Note I said intelligible. Many have offered me descriptions and attributes they ascribed to god, but even a cursory examination reveals internal contradictions. I doubt your discussion will pass this point in the debate. 

The obvious fallacy at the start is that of circular reasoning.

You can not prove God exist or does not exist. It is and exercise in futility

For a religious person it becomes circular reasoning.

For the non-religious person it becomes an argument from silence.

The first step is to realize it's an exercise in futility.

The most productive way to salvage your time and effort is to utilize the engagement to sharpen your own debate skills and potentially inspire someone else who reads or overhears the discussion to be more outspoken or better informed. Beyond that, you're basically wasting your time, and that's okay if you perfect your own strategies in the interim.

Don't be attached or deluded into thinking you will convert them. It's possible but highly unlikely. At best, think of your effort as planting a grape seed that might in time work its way into their defense systems and thought patterns and ultimately agitate for clarification, reason, and enlightenment.  

The biggest mistake is thinking you are arguing with a rational human being. There's a big difference between learned ratiocination and biased rationalizing, as Dr. Bo points out.

In fact, there are so many fascinating cognitive and psychological factors involved there are literally dozens of different areas of study and discipline with dozens and dozens of names, categories, and definitions for the condition: Cognitive bias, Identity-protective reasoning, psychological conditioning, cultural imprinting, brainwashing, religious indoctrination, ideological brain viruses, memes, memoriter, promiscuous teleology, hyperactive agency detection, etcetera. 

Steven Pinker explains it most definitively:

"Challenge a person’s beliefs, and you challenge his dignity, standing, and power. And when those beliefs are based on nothing but faith, they are chronically fragile. No one gets upset about the belief that rocks fall down as opposed to up, because all sane people can see it with their own eyes. Not so for the belief that babies are born with original sin or that God exists in three persons or that Ali is the second-most divinely inspired man after Muhammad. When people organize their lives around these beliefs, and then learn of other people who seem to be doing just fine without them–or worse, who credibly rebut them–they are in danger of looking like fools. Since one cannot defend a belief based on faith by persuading skeptics it is true, the faithful are apt to react to unbelief with rage, and may try to eliminate that affront to everything that makes their lives meaningful." 

Another helpful observation is a favorite quotation from the brilliant screenwriter Robert Oxton Bolt, who opined:  "A belief is not an idea the mind possesses, it is an idea that possesses the mind."

Disclaimer: This is not to say that sometimes the whole endeavor can be hilariously entertaining if not a teensy bit sadistic and cruel. And that's okay sometimes too. :)

Interestingly enough Thomas Jefferson wrote of his frustration in challenging Trinitarians by admitting: Ridicule is the only weapon which can be used against unintelligible propositions. (Letter to Francis Adrian Van der Kemp (30 July 1816)

Theoretical physicist Lawrence Krauss had giddily pronounced at times:  Ridicule is fun!

It can be constructive to have a reserve of pithy Zen-like retorts to the most ridiculous claims of religionists. And if properly delivered it may be the very thing that eventually shocks them out of their intellectual stupor.

Your question seems to be more about an argumentation strategy than a fallacy. Although we try to focus on fallacies here, I can say that people who offer up fallacious reasoning to justify their positions are often just rationalizing , that is, attempting to give a reason even when the reason given is not the real reason. A good strategy is to begin with asking if that is the reason they believe in God, and ask if you clearly demonstrate that to be a poor reason, would they stop believing in God.

You have correctly identified her points as circular reasoning and begging the question .  As such, the reasoning is fallacious.  However, having flawed logic or poor reasoning in itself does't make the conclusion false.  There may or may not be alternate arguments or lines of reasoning that do lead to a valid conclusion that is the same one she has reached already through the circular reasoning.

Even if you were able to demonstrate to your discussion partner that her line reasoning doesn't prove the existence of a particular God, you probably won't dissuade her of her belief in the God.  Her conclusion is just that - a belief; it's not something derived logically from her circular argument.  And that's totally OK.  If your goal is to change her belief, logic probably won't do the trick – unless you've got an iron-clad, fully-logical and accepted proof of the existence of God hiding in the wings.