I am part of an English-speaking community that engages in serious competitive quizzing. Most of the quizzing is done in-person, but occasionally, more casual quizzes are conducted online (for convenience's sake), via Skype or instant messaging, and in uncontrolled environments (i.e. the participants connect from their home computers and aren't overseen by proctors or anything), and so this of course entails a risk that the participants will Google for answers as they participate. Indeed, this has happened to us more than once.
Despite this, we do not have a term for this behavior; when we do want to refer to it by some shorthand, we usually use some locution involving the name of the perpetrator of the most notorious cheating-by-Googling incident, e.g. "pulling a Fred" (except that the name of the perpetrator is not Fred).
All of this is to say - Dave Magner's answer is correct; there is no term for this. But hey, if you want, check back with me in a decade or so - I suspect by then there'll have been enough instances of this that we'll have coined a term for this behavior.