You can use the Magento\Quote\Model\QuoteRepository class to get the quote subtotal in both the front and backend. This class is used to retrieve quotes from the database and will have the quote subtotal included.
You can call this class in the front end using the \Magento\Framework\App\ObjectManager.
In the backend, you can call it using the Magento\Quote\Model\QuoteRepositoryFactory.
Example:
//Frontend
$objectManager = \Magento\Framework\App\ObjectManager::getInstance();
$quoteRepository = $objectManager->create('Magento\Quote\Model\QuoteRepository');
$quote = $quoteRepository->get($quoteId);
$subtotal = $quote->getBaseSubtotal();
//Backend
$quoteRepositoryFactory = $this->_objectManager->create('Magento\Quote\Model\QuoteRepositoryFactory');
$quoteRepository = $quoteRepositoryFactory->create();
$quote = $quoteRepository->get($quoteId);
$subtotal = $quote->getBaseSubtotal();