I do...I no longer tell people Google it, I now say Alphabet it and once I feel they are sufficiently trained I simply say bet it. Try it and watch it catch on to the people around you.
Until people search things at alphabet.com instead of google.com I suspect you will be fighting an uphill battle to change this particular bit of terminology.
I never even thought to check alphabet.com until your post, surprised to find it wasn't them. I am not to concerned about changing the vernacular, but looking at the history of Google as a verb, it looks like Larry Page first did in 1998, but it didn't become popular until a Buffy the Vampire episode in 2002. Also looks like out of concern Google might be generizied (lose its trademark protection) in 2006 Google asked the public to only use Google to refer to the company/services.