Yes, partially. In fact, that is (probably) why Google offers DNS servers for public use.
Such a DNS server would not know all the information about you, though. It knows:
- The immediate upstream server that the request came from
- The FQDN of the server that you are trying to visit (e.g., security.stackexchange.com)
- The destination domain of emails (from MX record lookups)
- In some cases also what other activities you might be engaging with that server (for instance, if your computer tries to look up an SSHFP record, it is safe to assume that you are going to use SSH).
- It may also be possible (I don't know for sure) to identify what type of device requested a particular DNS name.
It does not know:
* Where the request actually originated (i.e., if you use your ISPs DNS server, and your ISP uses Google as a forwarder, then Google would only see your ISP's DNS server, not your personal computer)
* The rest of the URL, cookies, etc. In some cases, knowing other domains that were requested at the same time may disclose some information.
* How often you visit a particular site within a few minutes. Generally, your computer will request a domain only once, and then keep the information cached for a while. So the DNS server only sees one request for security.stackexchange.com even if you browse a dozen pages on that site.
Bottom line: some tracking is possible, but it will not be comprehensive and probably of little direct use to advertisers. It may be very valuable for state actors, though - simply knowing that somebody looked up the domain "www.wikileaks.com" or "www.howtojointhejihad.com" can be interesting.
For a company like Google, the value is probably more in the aggregate - their DNS servers give them an idea which Web sites are most popular, and may also tell them about Web sites that Google's crawlers haven't discovered yet.