post https://{baseUrl}/V1/storeview//user/quote
Use this to create a new quote. You can use the id provided in the successful response body to add/remove/update items in your quote.
Sample Curl
curl --location --request POST 'https://mygatewayurl.com/V1/storeview/default/user/quote' --header 'Authorization: Bearer qsk2sdasdlkcgsuhhkbale7vk571vn'