Add a custom meta box to the new/edit category pages. The meta data is saved using update_term_meta.

term-meta.php

<?php

/**
 *Add meta box to the term category page.
 */
function dtbaker_taxonomy_edit_meta_field( $term ) {
	// Retrieve the existing value(s) for this meta field.
	$term_meta = $term && !empty($term->term_id) ? get_term_meta( $term->term_id, 'custom_page_id', true ) : false;
	?>
	<tr class="form-field">
		<th scope="row" valign="top"><label
				for="term_meta[custom_page_id]"><?php _e( 'Choose a page', 'dtbaker' ); ?></label></th>
		<td>
		<?php
		$dropdown_args = array(
			'post_type'        => 'page',
			'selected'         => $term_meta,
			'name'             => 'term_meta[custom_page_id]',
			'show_option_none' => __( '(no page)' ),
			'sort_column'      => 'menu_order, post_title',
			'echo'             => 1,
		);
		wp_dropdown_pages( $dropdown_args );
		?>
			<p class="description"><?php _e( 'Choose a page', 'dtbaker' ); ?></p>
		<?php wp_nonce_field( 'update_term_meta', 'term_meta_nonce' ) ?>
		</td>
	</tr>
	<?php
}

add_action( 'category_add_form_fields', 'dtbaker_taxonomy_edit_meta_field', 10 );
add_action( 'category_edit_form_fields', 'dtbaker_taxonomy_edit_meta_field', 10 );

/**
 * Save meta data callback function.
 */
function dtbaker_save_taxonomy_custom_meta( $term_id ) {
	if (
		isset( $_POST['term_meta'] ) && is_array( $_POST['term_meta'] ) &&
		! empty( $_POST['term_meta_nonce'] ) && wp_verify_nonce( $_POST['term_meta_nonce'], 'update_term_meta' )
	) {
		foreach ( $_POST['term_meta'] as $key => $value ) {
			update_term_meta( $term_id, $key, sanitize_text_field( $value ) );
		}
	}
}

add_action( 'edited_category', 'dtbaker_save_taxonomy_custom_meta', 10, 2 );
add_action( 'create_category', 'dtbaker_save_taxonomy_custom_meta', 10, 2 );