The "unread" indicator behaves in the same way when using the quire-api/browser or mobile clients. It shows when a new task or comment was added by another user, through either UI or api.
Neither the UI nor the quire API have a function to explicitly control this status indicator.
So as of now it would not just be a "new/missing" api feature first, it's rather a new completely new Quire-Feature, which had to be implemented first. You can post your feedback and ideas in the Quire Feedback project (keyword 'orange dot') or send an email to info@quire.io / support@quire.io.
So as of now you can trigger this by adding a comment or add a new task via API.