I’m afraid no one-word answer to this question could be accepted as a valid French verb. Any applicable one would be nothing but convenient jargon. I have heard googliser as well as googler. A proper way of saying that would simply be (re)chercher sur Google.

In English though, I thought “to googlize” was more commonly used as “to google” for this meaning, but it might simply not be the case anymore. It’s just a matter of fashion in my opinion…

Google is a trademark for a popular search engine (moteur de recherche : Franceterme, Termium, GDT). Only Termium lists the verb based thereupon and provides a translation, “googler“; incidentally it also lists the googler (noun), the person who uses the search engine to find information on the Internet, and provides “googler/googleur(Fr.)” in that case.¹ The verb has certainly been used in French.

Collins suggests “chercher + sur/avec” as suggested in another answer. It also covers this typical scenario where the search term is explicitly stated within the sentence using the verb “taper” : “J’ai tapé `vols bon marché pour Mumbai’ dans Google.” (Collins). These verbs are very usual in context.

_{1. It’s interesting to look back on some articles from 10-15 years ago on the topic; an article (2003) references concerns for the “dégénérescence de la marque” when used casually as a noun etc.. Another article (2006) claims Google recommended against using their name as a verb and preferred « effectuer une recherche sur Internet avec Google ». Please note the answer takes no stance on whether the verb to google means to google with Google or to search with any search engine in English; the AmHDoEL entry didn’t specify and it is off topic. The point is just Termium has identified some sort of meaningful use in French and connected it to the English form. Franceterme, GDT and Larousse don’t have the verb: they have the “moteur de recherche“. “Googler” yields no ngrams from the French corpus.}

You may use “Google-le” when talking about a specific subject. Otherwise there is an expression: “Google est ton ami” : “Google is your friend” to obviously suggest someone to do his own research…

One would say “chercher sur Google” or “faire une recherche sur Google” but one can expect that “gougler” would probably be understood even though it would sound a bit strange as it has not been officially frenchified yet.