I was just about to say the same thing. Thomas is the first person who comes to mind. He said he wouldn't believe Jesus rose from the dead unless he saw him and his scars. Job is an entire book about wrestling with God. It's all about why would a good God allow all this suffering? All his friends tell him to just denounce his faith and move on with life. And several of the people throughout the Bible don't doubt that God exists, but they do doubt that He will do what He promises.

So there are definitely religions that encourage doubting whether God exists. Eventually you have to come to some sort of immovable mover. Whether that's the Big Bang or God, so there's nothing inherently illogical about believing in something that is timeless and has always existed.