If you read the bible, God was not a mysterious person at all. He would talk to people and he would even negotiate. In the story of Sodom and Gomorrah, some guy was arguing with God over how many good people must live in those towns before God would spare them. If that God exists, we shouldn't even be looking for evidence of him. He would be as easy to notice as the blue penis man from Watchmen.
That "some guy" was a prophet named Abraham who was considered on of the greatest prophets of the scriptures of three major faiths, Christianity, Judaism, and Islam. The dealings of the bible were with people that were prophets and in direct communication with God. Most people mentioned in the bible did not enjoy this relationship.
Your perception of "that God" makes many assumptions and still lacks any parameter that could constitute anything approaching research of the subject.
Science is a disciple. Not a backyard hobby where you put things together as see what happens. if you're lucky by doing that you can approach the scientific age of the early 1900's.
As it has been pointed out there are many religions with Gods. So take the most basic parameters for a god like being and tell me what kind of evidence such a being would leave behind?
Intentionally or unintentionally?
So say that he would make himself known and famous is to assume a trait of his personality to make himself publicly known, such a question best fielded by a psycologist, not a scientist.
Unintentionally left evidence is get more into the sciences such as forensic evidence.
IF two men, non gods had a conversation in a wooded area, after a month of growth and animal movement what evidence would there be that the conversation took place?
I think reading people's comments online makes people that want to be scientific lazy. Too many scientific types are taking people's words for granted without inquiring themselves.
Welcome to the age of information. Its easy to grab others claims as assume them as your own because it feels fun or popular. Real minds examine, not assume.