Summary section created on checkout-module.
You can change checkout_index_index.xml, add item as 'quote' & create js for
Magento_Checkout/js/view/summary/quote
Path - \vendor\magento\module-checkout\view\frontend\layout\checkout_index_index.xml
<item name="quote" xsi:type="array">
<item name="component" xsi:type="string">Magento_Checkout/js/view/summary/quote</item>
<item name="config" xsi:type="array">
<item name="title" xsi:type="string" translate="true">Quote</item>
</item>
</item>
\vendor\magento\module-checkout\view\frontend\web\js\view\summary
Also create template file in below path. vendor\magento\module-checkout\view\frontend\web\template\summary
You can override this file on you custom extension or update core files (Not recommended). This might not be exact answer but might be workaround to you. Hope this help you.