A montage scene of newspaper headlines early on has March 12, 1939 as the date. Much later in the film, Jude Law asks Gwyneth Paltrow the date so he can calculate the position of a star. She responds "March 2" but there is absolutely nothing to indicate that nearly a year has passed.
After the robots first attack New York City, a series of newspapers is shown. On the Japanese one, you can see Godzilla's outline.