remove everything from wp_nav_menu

You can use another walker than WordPress pre-build one.

Add this to your functions.php file in the theme:

class Clean_Nav_Walker extends Walker {
    var $tree_type = array( 'post_type', 'taxonomy', 'custom' );
    var $db_fields = array( 'parent' => 'menu_item_parent', 'id' => 'db_id' );
    function start_lvl(&$output, $depth) {
        $indent = str_repeat("\t", $depth);
        $output .= "\n$indent<ul class=\"sub-menu\">\n";
    }
    function end_lvl(&$output, $depth) {
        $indent = str_repeat("\t", $depth);
        $output .= "$indent</ul>\n";
    }
    function start_el(&$output, $item, $depth, $args) {
        global $wp_query;
        $indent = ( $depth ) ? str_repeat( "\t", $depth ) : '';
        $class_names = $value="";
        $classes = empty( $item->classes ) ? array() : (array) $item->classes;
        $classes = in_array( 'active', $classes ) ? array( 'active' ) : array();
        $class_names = join( ' ', apply_filters( 'nav_menu_css_class', array_filter( $classes ), $item, $args ) );
        $class_names = strlen( trim( $class_names ) ) > 0 ? ' class="' . esc_attr( $class_names ) . '"' : '';
        $id = apply_filters( 'nav_menu_item_id', '', $item, $args );
        $id = strlen( $id ) ? ' id="' . esc_attr( $id ) . '"' : '';
        $output .= $indent . '<li' . $id . $value . $class_names .'>';
        $attributes  = ! empty( $item->attr_title ) ? ' title="'  . esc_attr( $item->attr_title ) .'"' : '';
        $attributes .= ! empty( $item->target )     ? ' target="' . esc_attr( $item->target     ) .'"' : '';
        $attributes .= ! empty( $item->xfn )        ? ' rel="'    . esc_attr( $item->xfn        ) .'"' : '';
        $attributes .= ! empty( $item->url )        ? ' href="'   . esc_attr( $item->url        ) .'"' : '';
        $item_output = $args->before;
        $item_output .= '<a'. $attributes .'>';
        $item_output .= $args->link_before . apply_filters( 'the_title', $item->title, $item->ID ) . $args->link_after;
        $item_output .= '</a>';
        $item_output .= $args->after;
        $output .= apply_filters( 'walker_nav_menu_start_el', $item_output, $item, $depth, $args );
    }
    function end_el(&$output, $item, $depth) {
        $output .= "</li>\n";
    }
}

Then make sure to specify the new custom Walker when calling wp_nav_menu().

<?php
  $walker = new Clean_Nav_Walker();
  $attr = array( 'theme_location' => 'quicklink', 'walker' => $walker);

  wp_nav_menu( $attr );
?>