global $wp_query;
$category_name = $wp_query->query_vars['product_cat'];
$category_object = get_term_by('name', $category_name, 'product_cat');
$category_id = $category_object->term_id;

$cate = get_queried_object();
$cateID = $cate->term_id;
echo $cateID;

<?php
global $post;
$terms = get_the_terms( $post->ID, 'product_cat' );
foreach ( $terms as $term ) {
    $product_cat_id = $term->term_id;
    if( $term = get_term_by( 'id', $product_cat_id, 'product_cat' ) ){echo $term->name;}
    break;
}?>