Everytime I close google, my steam account logs off

Try to check the site setting of Your Chrome in
Settings -> Advanced -> Privacy and Security -> Site Settings -> Cookies and site data -> Clear cookies and site data when you quit Chrome

make sure it's turned off

Also check what kind of extensions you are running with chrome. There are some that clear cookies everytime you close the browser and your steam browser session is stored in such a cookie.
For example i use Vanilla Cookie Manager which deletes all cookies that are not on the whitelist. If this extension is configured wrong it deletes your login session.

i had a similar problem
found somewhere that if you go to https://store.steampowered.com/twofactor/manage
and "deauthorize all computers", it fix the problem
it did for me for about a week.
never bothered to try again
hope it helps

Well, you haven't given much information about your actual usage, settings, etc...
So my first question would be: Did you choose " Remember me on this computer"
Second question would be: Do you use multiple computers?
This is the single one issue I have - when I login on another computer, it logs me out of the first computer I was in - this appears to happen irrespective of whether I've "remembered" the previous computer or not.

I had this problem while having certain extensions. eg. Hola made it happen with steam every time