In grammar, the term "proper noun" refers to the names of people, places and anything that is a sole thing of it's kind such as the seasons or cardinal directions. A proper noun is spelt with a capitalized first letter: Jack and Jill Smith, Washington, Summer, North etc.
In English grammar, we do not have "proper" verbs.
What you're really asking is, "can the word google be correctly used as a verb?"
The answer is yes.
Language changes all the time. Some words drop out of use and become archaic or obsolete.
New words join the language to meet new needs.
Because Google is one of the largest and most used search engines on the internet, we started using the word google as a verb to refer to researching or looking up information online.
"I googled it."
"You can google it anytime you like."
"Just because he googled it doesn't mean it's true; it depends on how reliable the source of the information was."