How can I use jQuery in Greasemonkey?

Perhaps you don't have a recent enough version of Greasemonkey. It was version 0.8 that added @require.

// @require https://ajax.googleapis.com/ajax/libs/jquery/3.4.1/jquery.min.js 

If you don't have 0.8, then use the technique Joan Piedra describes for manually adding a script element to the page.

Between version 0.8 and 0.9, @require is only processed when the script is first installed. If you change the list of required scripts, you need to uninstall your script and reinstall it; Greasemonkey downloads the required script once at installation and uses a cached copy thereafter.

As of 0.9, Greasemonkey behavior has changed (to address a tangentially related issue) so that it now loads the required scripts after every edit; reinstalling the script is no longer necessary.

There's absolutely nothing wrong with including the entirety of jQuery within your Greasemonkey script. Just take the source, and place it at the top of your user script. No need to make a script tag, since you're already executing JavaScript!

The user only downloads the script once anyways, so size of script is not a big concern. In addition, if you ever want your Greasemonkey script to work in non-GM environments (such as Opera's GM-esque user scripts, or Greasekit on Safari), it'll help not to use GM-unique constructs such as @require.

You can create a new script using the New User Script in Greasemonkey but you have to edit the config.xml file inside the gm_scripts folder.

Your config.xml file should have a similar syntax as this:

<Script filename="jquery_test.user.js" name="jQuery Test" namespace="http://www.example.com/jQueryPlay/" description="Just a test" enabled="true" basedir="jquery_test">
        <Include>http://*</Include>
        <Require filename="jquery.js"/>
</Script>

Notice the <Require> tag.

In your script you can use direct jQuery syntax. Make sure you have the require tag in the Greasemonkey header. Here is a Hello World example:

// ==UserScript==
// @name           Test jQuery
// @namespace      http://www.example.com/jQueryPlay/
// @description    Just a test
// @include        http://*
// @require       http://ajax.googleapis.com/ajax/libs/jquery/1.2.6/jquery.js
// ==/UserScript==

$(document).ready(function() {  
        alert("Hello world!");
});

Remember that after modifying the config.xml you have to restart your browser for Greasemonkey to reload the settings again.

Also note that you need to copy the jquery.js file to your script directory folder in order for this to work. I tested this, and it only works if you actually copy the file manually there.

Happy jQuerying!