Not all intelligence has to work the way we do, with a physical brain, synapse and electrical impulses, there are no rules saying that this is required for intelligence. What is required is communication between elements that create intelligence and we know that there is quantum entanglement / communication and that quantum particles exist in every possible scenario of existence, they are multidimensional they can travel and communicate instantly with no regard to SpaceTime.  The coalescence of an intelligence  based in quantum communication would be infinite and timeless. 

Not one individual synapse or electrical impulse is a thought, it is the coalescence of all input that create thought, for us it happens in a blink of an eye but for a quantum intelligence with no regard to SpaceTime the communication would be infinite as it would also be instantaneous because that is how quantum communication works. Instantaneous communication on an infinite scale leads me to the possibility that an infinite intelligence is not an unreasonable hypothesis that the coalescence of infinite communication would manifest as infinite intelligence. This is an extremely simplified concept there are many more types of energy in existence that are in play but the main thing to remember is consistency and communication.

It does not matter if you believe in God or not. The creation of our universe and all of existence was and is a natural phenomenon regardless whether we fully understand it or not, it was not magic or supernatural. The creation of our universe required infinite power, infinite energy, infinite force and the two most important things of all infinite focus and consistency. Everything that was required to create our universe and all of reality still exist, it still flows through all of creation. The question is was our universe a function of evolution, of spontaneous manifestation or creation? The first two do not require consciousness, intelligence, forethought, imagination and purpose of choice but the third does. Creation can only be manifested by a living self-aware intelligent consciousness. There is no creation without purpose. I believe our universe is a combination of all three. It is the creation of spontaneous manifestation through evolution. All of existence is the infinite intelligence/consciousness fulfilling It's Self, there can be no creation, no expression without manifestation. The greatest expression there could be from an infinite consciousness would be the manifestation of less than infinite beings, But with nearly infinite possibilities, through the process of evolution capable of original thought and imagination, beings that can truly create for and by themselves. Beings of choice that can evolve through their choices. Beings although less than infinite would be in the image and likeness of the infinite.

The coalescence of this instant communication and information IS infinite consciousness, encompassing and manifesting all of space and time.