This is an apologetics argument that I see from time to time. When Christians use this argument, they go heavy on the obfuscation: lots of big words and complex sentence structure in an effort to overwhelm you so that you feel too intimidated to offer a counterargument.
They will explain Gödel's Incompleteness Theorem in the most obtuse, unnecessarily complex way possible because if you actually understand the argument being made, it's very simple to pick apart. The solution, therefore is to find educational material that explains the Incompleteness Theorem from anyone other than theistic apologists. If the goal is education instead of dishonest obfuscation, you'll have an easier time understand what the theorem is saying, and thus see what is wrong with Christian apolegetics arguments based on it.
Aside: Mathematical Truths are Axiomatic
One thing you quickly understand from reading the Wikipedia article on the Incompleteness Theorem is that math truths are axiomatic. This is something that far too many people gloss over in discussions about mathematical truths.
If you ask a modern mathematician "where does math come from," they will point you to set theory because you can use set theory to derive nearly everything else. Of course, set theory comes from the 1870s. When most people ask the above question, they are really inquiring about the origin of math. Arguably, math comes from counting. Once you have a counting system, addition is just adding twice. Subtraction is the reverse of adding. Multiplying is adding where the number of times you add something depends on the numbers being multiplied. From there, you can branch out to all the other stuff.
Whether you start from counting or start from set theory, mathematical truths are true because they are internally consistent with a set of rules and definitions that we all agree to at the onset. If you do not agree to all of the rules and definitions, then certain (or all) math truths will not be true for you.
Thus, math truths are like the statement "green means go, red means stop." That statement is true only if we agree to follow the set of rules called "traffic laws." If someone does not agree to those rules, then the above statement is not necessarily true. Math truths cannot and do not exist independent of certain basic rules and definitions, and indeed, exist only because of those rules and definitions. As a former physics major, I would love to tell you that math is some kind of intrinsic truth, but it's not.
Thus a mathematical proof is not proving a math concept to be true in the sense most people think when we talk about truth, but rather a mathematical proof merely demonstrates that a math concept is internally consistent with the set of rules and definitions we all agree to when we do math.
Extremely Reductive Version of Gödel's Incompleteness Theorem
I would invite you to watch the YouTube video linked above, or if you prefer, search the internet for some other educational material explaining Gödel's Incompleteness Theorem. Put another way, there are things that are true in math that we will never be able to prove. This is why the Incompleteness Theorem caused mathematicians to freak out.
Application to Apologetics
Hopefully, you can immediately see problems with any attempt to apply this to arguments for the existence of a deity or deities.
For one thing, this really only applies to mathematics, or any system of truth that restricts itself to using the same tools as mathematics for supporting truth claims, and further restricts itself to axiomatic truths. Simply by applying Gödel's Theorem to a claim about the existence of a deity makes that truth claim axiomatic, in which case, why bother having a discussion about this at all since it is obvious that the theist is going to arbitrarily select rules and definitions that ultimately lead to the conclusion that a deity exists.
You can define math into existence because math is purely an idea. It is an idea that just happens to be a very useful tool for describing the world around us, but it is still just an idea, and it is an idea that has no existence except to sentient minds that agree to accept certain rules and definitions. When theists insist that their god or gods exist, that is not the kind of existence they are talking about.
Another tactic is to point out that the Incompleteness Theorem implies an "outside observer," since only an omniscient being could possibly know those unprovable truths. This is of course absurd. It's more than possible that no one knows those truths and no one ever will. The fact that we wish we knew those truths does not mean that someone exists who does. This is purely wishful thinking.
The weakest version of this argument is to point out that since there are math truths that can never be proved, and that since God has never been proved, God therefore exists. This version of the argument could be applied to every claim that has never been proved, and would force us to accept a large number of absurd and blatantly false claims were the logic of this argument valid.
Don't Fall for Obfuscation
Once you understand the Incompleteness Theorem, the apologetics arguments that lean on it fall apart under any amount of examination. This is why Christian apologists work so hard to make the Incompleteness Theorem as incomprehensible as possible. The less you understand about the argument being made, the harder it will be for you to pick it apart. The solution, therefore, is to simply seek out educational material that discusses the Incompleteness Theorem without regard to apologetics discussions, since an educator's goal is to help you understand, while the Christian apologists' goal is to prevent your understanding.