I tried to use the list thread method:
GET https://www.googleapis.com/gmail/v1/users/{MyEmailAddress}/threads?labelIds=INBOX&labelIds=UNREAD&key={YOUR_AP
GET https://www.googleapis.com/gmail/v1/users/{email}/messages?labelIds=UNREAD&labelIds=INBOX&fields=messages%2FthreadId
"messages/threadId" will count the number of unread threads but will list the same thread multiple times if the thread has multiple messages. Store them in an array and check the array as you fill it for matching "threadId" identifiers and ignore duplicates.