SoulKeeper Although, I am not using Google search engine but can Google collect search data
Google cannot collect your local browsing history, however If you use Google search in the said browser; Google will collect data of 'Google search'—and It will also link data of Google search to your account, Unless you sign out (Which will sign you out of all Google properties, not just G-Mail or Google search), use Google search in an incognito tab in the said browser or use Google search in an another browser without signing-in.