Chrome has had DNS over HTTPS support available since February of 2018, however, Google has had the feature disabled as they work on the UI and make sure this doesn't cause deployment issues with the many environments where Chrome runs (schools, enterprises, etc).

As per https://crbug.com/799753#c8 (comment 8), you can enable DOH on Chrome prematurely by appending command line options when starting Chrome.

To enable this on Windows, right click Chrome on your taskbar, then right click "Google Chrome" again and select "properties".

Now you'll see information about the Chrome shortcut. In the "target" text box, replace the entire box with the following text:

chrome.exe --enable-features="dns-over-https<DoHTrial" --force-fieldtrials="DoHTrial/Group1" --force-fieldtrial-params="DoHTrial.Group1:server/https%3A%2F%2F1.1.1.1%2Fdns-query/method/POST

Finally, restart Chrome. This enables the feature and sets the DoH resolver to Cloudflare's 1.1.1.1 resolver. If you also launch Chrome via the desktop icon, perform the same actions on that desktop shortcut.

To confirm DNS over HTTPS is set up, you can visit 1.1.1.1/help and "Using DNS over HTTPS" should be a "Yes".

If you use a different operating system, see this chromium wiki post for instructions on starting Chrome with command line options. Use these flags:

--enable-features="dns-over-https<DoHTrial" --force-fieldtrials="DoHTrial/Group1" --force-fieldtrial-params="DoHTrial.Group1:server/https%3A%2F%2F1.1.1.1%2Fdns-query/method/POST