Update custom column data in quote table Magento 2

The easiest way to solve this is to fill in this field in the quote model and save it using the Repository:

class QuoteUpdater
{
    /**
     * @var \Magento\Quote\Model\QuoteRepository
     */
    protected $quoteRepository;

    /**
     * @param \Magento\Quote\Model\QuoteRepository $quoteRepository
     */
    public function __construct(
        \Magento\Quote\Model\QuoteRepository $quoteRepository
    ) {
        $this->quoteRepository = $quoteRepository;
    }

    public function updateQuoteData($quoteId, int $customData)
    {
        $quote = $this->quoteRepository->get($quoteId); // Get quote by id
        $quote->setData('custom_column', $customData); // Fill data
        $this->quoteRepository->save($quote); // Save quote
    }
}

To set data in quote table, use below code -

$objectManager = \Magento\Framework\App\ObjectManager::getInstance();
$quoteId = 10;
$customColumnValue = 1; 
$quote = $objectManager->create('Magento\Quote\Model\Quote')->load($quoteId);
$quote->setCustomColumn($customColumnValue);
$quote->save();

Here replace $quoteId variable with your quote ID .