What is the name of field that stores page visits count data? or how i can get data about visits?
If you want visits on the search interface specifically, you could use the Coveo Cloud Administration Platform, in the Reports section, there should be a Dashboard that is called Summary. It should give you enough information about visitors of your search interface.
If you want visits from your whole site, this is what we call Page Views.
I suppose you are using Coveo for Sitecore, which provides a Sublayout and a View for that purpose.
Insert the Coveo Page View Analytics component in a way that it will be included in all of your pages. If you have some kind of "header" mechanic that is applied to every page, this is where you should include it.
You can refer to the Logging Page View Events page for more information.