Typing in Godzilla or any other Kaiju like Mothra, King Ghidorah, Anguirus, Rodan, Mechagodzilla, King Caesar, Zilla, Titanosaurus or even King Kong will give you "System cannot confirm or deny the existence of this lifeform." In other words, yes.
If you try to cheat and type in "Show Trailer" or "Spoilers" it will respond with "Did you really think it would be that easy?"
Typing in "Roar" will produce "SKREEONK!"
Typing in "IhateGodzilla" will produce "The data you entered is incorrect."
"Knock Knock" produces "M.U.T.O. already knows who is there."
"Say something" or "Say my Name" - “Something. Please stop wasting M.U.T.O. resources."
This Huffington Post interview with Gareth Edwards has this funny anecdote about the design team:
Gareth Edwards: You'd call people and say, "Would you work on the film with us?" And they would say, "Aw, I'd like to, but I'm really busy." And then you'd say, "It's Godzilla," and they'd go, "When do you need me?"
The day after the second trailer arrived, Godzilla, Rodan and King Kong were on the front page for Bing's image search, in the "Animals" section. Yes, right alongside normal animals like giraffes and elk.
This Snickers commercial made to promote the movie. Apparently, Godzilla constantly goes on rampages whenever he hasn't had his Snickers.
The Facebook Q&A with Edwards following the 2014 SXSW event has a few:
When asked about sequel ideas:
Barron Christopher: "Have you already started gathering ideas for a sequel? How many Godzilla films would you like to do?
Edwards: "I feel like I'm about to give birth, so the last thing I want to do is get back into bed, but I'm sure once I see the child, I'll be racing to have another."
When asked if King Ghidorah would appear in sequels, Edwards treats him like an actual actor:
Edwards: "We're contacting his agent. They're asking for a lot of money."
A similar one:
Robert John Mack II: "Can Godzuki be in the movie?"
Edwards: "We auditioned him, but there's all kinds of issues with child labor laws."