Plugin Settings Color Input Javascript Event Listeners
If this is your plugin, you can register an AssetBundle with custom CSS and JS for your plugin. This will be loaded in your plugin settings page, and in your JS file you can then specify an event handler for your colour picker. You might need to add a specific class to the input in the settings template to ensure you can select it easily in your JS file.
If this is referring to a first/third-party plugin, you could use a plugin like CP JS to load a custom JS file into the CP and attach an event handler in that file to trigger your XHR.