Try one of these extensions:

• Auto Cursor: https://addons.mozilla.org/firefox/addon/auto-cursor/
• Custom New Tab: https://addons.mozilla.org/firefox/addon/custom-new-tab/

If I make https://www.google.com/ my home page, the cursor does move into the search box -- when it is loaded as my home page in a new window (Ctrl+n) or using the Home button.

However, as cor-el's reply suggests, it is different if you are using the Google page as a substitute new tab page. The new tab page code overrides the normal behavior of the Google home page. If that is where you are encountering the problem, one of those add-ons should help.

If you install Custom New Tab, you should disable New Tab Homepage (or any other extension you are using to make Google your new tab page) to avoid conflicts.

After installing Custom New Tab, you need to use its Options page to set the desired new tab page (instead of using about:config).

Open the Add-ons page using either:

• Ctrl+Shift+a (Mac: Cmd+Shift+a)
• "3-bar" menu button (or Tools menu) > Add-ons

In the left column, click Extensions. Then on the right side, find Custom New Tab and click its Options button.

Entering your preferred home page address is simple. If you do not check either of the boxes, the extension will set the focus in the page so, for example, the cursor will be in the Google search box. (See screen shot attached.)

I had to exit Firefox and start it up again before Custom New Tab started working, but this may have been due to testing so many extensions back-to-back.

Success?