How to correctly load this jquery script through the file functions.php?

You don’t need to include jquery-ui and jquery-ui.min either of them is sufficient . Also jquery and jquery-ui are already included in WordPress.

Try this 

wp_enqueue_script( 'jquery');   
wp_enqueue_script( 'jquery-ui-core');   
wp_enqueue_script( 'qs.slider', get_template_directory_uri() . '/layout/assets/js/qs.slider.js', array('jquery','jquery-ui-core'), '', true );    
wp_enqueue_script( 'qs.slider.init', get_template_directory_uri() . '/layout/assets/js/qs.slider.init.js', array('jquery','jquery-ui-core'), '', true );

The above code will make sure jquery and jquery-ui is loaded before loading your slider.

Load the styles as you’re already doing.

Please note that only loading these files will not do anything on its own you need to actually use the HTML markup provided by the author of this slider, refer to this source code. And customize the slider as per your needs and then add to your custom page.

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