I might answer “do you think Google does X” with ‘yes’ because I believe they do, and “do you expect Google to do X” with ‘no’ because I do not believe it would be right to do so.

“Expect of” is often used to connote a kind of standard setting.

I expect power multinational companies to behave ethically to the best of their ability. I believe they behave ethically only to the degree that it allows them the political leeway to behave without scrutiny.