Yes, if the customer has enabled notifications for your website and closed it from browser then also they will get notifications if set in Magento 2 Web Push Notifications Extension.

Read this guide on How to Configure Push Notification in Magento 2 to setup this extension.

Here are the few more resource to help you get most out of this Magento 2 Push Notifications Extension.

• What is Push Notification? How is it Beneficial for an eCommerce Store.
• Push Notifications Magento 2 Extension - Send Web Updates to Browsers.