There is a GNOME Shell extension on GitHub called Web Search Provider for GNOME Shell that lets you add different web search providers.
Note that adding Extensions from non trusted sources is unsupported, not recommended and you may encounter stability and/or security issues.
Now if you still want to set it up, download the extension:
git clone --depth=1 https://github.com/mrakow/gnome-shell-web-search-provider ~/.local/share/gnome-shell/extensions/gnome-shell-web-search-provider@mrakow.github.com
Edit config.json file:
gedit ~/.local/share/gnome-shell/extensions/gnome-shell-web-search-provider@mrakow.github.com/config.json
And replace its content with the following:
{
"searchEngines": {
"Google Search": {
"urlTemplate": "https://www.google.com/search?q={searchTerms}",
"iconPath": "/usr/share/icons/gnome/256x256/actions/search.png"
}
}
}
Restart GNOME Shell: Alt+F2, r, then Enter. Or logout and login again (for Wayland).
Enable the Web Search Provider extension in GNOME Tweaks or at
https://extensions.gnome.org/local.
Now a Google Search entry should appear whenever you use GNOME Shell search.
Tested on Ubuntu 19.04