To change the social icons order you can simply paste this function in your child theme’s functions.php file and reorder the lines in the $profiles array to match the order you want for them:

function quadro_social_icons($area, $class, $style, $color) {

	// Retrieve Theme Options
	global $quadro_options;

	$icons_style = isset( $quadro_options[$style] ) ? esc_attr( $quadro_options[$style] ) : '';

	$profiles = array( 
		'dribbble' => 'Dribbble', 
		'facebook' => 'Facebook', 
		'flickr' => 'Flickr', 
		'github' => 'Github', 
		'google-plus' => 'Google+',
		'instagram' => 'Instagram', 
		'linkedin' => 'Linkedin', 
		'pinterest' => 'Pinterest', 
		'skype' => 'Skype', 
		'tumblr' => 'Tumblr', 
		'twitter' => 'Twitter', 
		'vimeo' => 'Vimeo', 
		'vimeo-square' => 'Vimeo', 
		'youtube' => 'Youtube', 
	);

	if ( $quadro_options[$area] == 'show' ) {

		echo '<ul class="social-area ' . $icons_style . ' ' . esc_attr($quadro_options[$color]) . '-color ' . $class . '">';

		foreach ( $profiles as $profile => $name ) {
			if ( !isset($quadro_options[$profile . '_profile']) || $quadro_options[$profile . '_profile'] == '' ) continue;
			echo '<li><a href="' . esc_url( $quadro_options[$profile . '_profile'] ) . '" target="' . esc_attr( $quadro_options['social_target'] ) . '" title="' . $name . '"><i class="fa fa-' . $profile . '"></i></a></li>';
		}

		echo '</ul>';

	}

}