The extent of Godzilla’s intelligence varies throughout the character’s history, but Godzilla is generally depicted as a thinking creature.
In the original Godzilla film and its early sequels he is depicted with a simplistic animal cunning, but as the Showa series progressed he is seen as being as intelligent as a human, capable of abstract thought, relating cause and effect.
Godzilla was shown to be able to figure out that his atomic breath could be used to power the electrodes needed to dry out Hedorah in Godzilla vs. Hedorah and able to communicate with other monsters. He realized that Mecha-King Ghidorah was being psychically controlled through several radio antennas and that destroying them would weaken the connection. He also possess a certain level of psychic power, possibly the most powerful in existence. His ability to track down enemies from far distances and constant evasion from sonar detection are implications of precognition.
In the Heisei series Godzilla reacts on animal cunning and instinct more consistently than in his Showa counterpart, as demonstrated by his conditioned response in The Return of Godzilla. He was still capable of independent thought, however and according to Miki Saegusa of human-like sentiments as well. This was corroborated by his mourning the death of Godzilla Junior in Godzilla vs. Destoroyah. In Godzilla vs Space Godzilla he was the first to figure out that Space Godzilla was using a tower at the central of a field of crystals to draw power several minutes before the JSDF was.
In the Millennium series, Godzilla’s behavior was again limited to a simplistic animal cunning. He was shown to be capable of abstract, even strategical thinking but displayed little to no anthropomorphism.
MonsterVerse Godzilla has shown good intelligence during the battle with the MUTOs. When being overpowered, he gradually learned their tactics during the battle and their weaknesses. After knowing male MUTO’s attack, he peaked around and used his tail to kill him.
Though the American-shot inserts from King Kong vs. Godzilla have little consistency with the timeline and rules of the Godzilla character in the Toho films, it may be worth mentioning that one of the American characters says that Godzilla has a brain about the size of a marble (“He’s sheer brute force”); in comparison, the character states that Kong “Is a thinking animal.” Again, this is largely inconsistent with Godzilla’s character.
Mechagodzilla II reveals that Godzilla has two brains, one one of them is located where his tails meets his torso. This brain is what controls Godzilla’s movement. Mechagodzilla was equipped with a harpoon-like weapon that could pierce Godzilla’s muscle tissue and destroy the brain by blasting it with electricity.
In more recent comics adventures, Godzilla is hypnotized and used by aliens to attack Earth’s cities.