If I understood correctly you need to show a specific category index on a page template, right?
If that’s the case you can use WP_Query:
<?php /* Template Name: My Page Template */ ?>
<?php get_header(); ?>
<!-- If you want to retrieve the page title, content, ... -->
<?php the_post(); // set up the post ?>
<h1> <?php the_title(); ?> </h1>
<div class="content"> <?php the_content(); ?> </div>
<!-- Then your category index -->
<?php
$category_id = get_cat_ID('YOUR CATEGORY NAME');
$args = array(
'cat' => $category_id,
'paged' => get_query_var('paged') ? get_query_var('paged') : 1
);
$cat_posts = new WP_query( $args );
if($cat_posts->have_posts()):
while($cat_posts->have_posts()):
$cat_posts->the_post();
?>
<div class="cat_post">
<h1> <?php the_title(); ?> </h1>
<div class="content"> <?php the_content(); ?> </div>
</div>
<?php endwhile; ?>
<div class="pagination">
<?php next_posts_link( __('Older Entries'), $cat_posts->max_num_pages ); ?>
<?php previous_posts_link( __('Next Entries') ); ?>
</div>
<?php wp_reset_postdata(); ?>
<?php endif; ?>
<?php get_footer(); ?>
Remember to change YOUR CATEGORY NAME accordingly.