Yes, it is possible to allow write access to multiple users.
QuickBlox released iOS SDK 1.7 with Permissions API, please check more details here: Custom Objects | iOS SDK | QuickBlox
The same you can find for Android SDK: Custom Objects | Android SDK | QuickBlox
REST API documentation can be found here: Custom Objects
You can easily grant access to records to your users, as well as specify what operations are allowed (READ, CREATE, UPDATE, DELETE).