This is a problem (
AFAIK).
Unfortunately, Google does not currently have a task-syncing API (i.e. it should support syncing via CalDav, but doesn't). Zimbra supports task-syncing via CalDav.
However, Google is considering introducing enhancements to Google tasks, including syncing, see here for latest news on this:
Top tasks for Google Tasks - Official Gmail Blog
In the meantime, if you have a Mac, you could try:
1) syncing tasks to iCal (if you are using a Mac) - you will probably need to install this:
iGTask -- Synchronize task between Google task and iCal Task
2) then re-syncing the same tasklist from iCal to Zimbra...
Alternatively, if you have an Android mobile device, you could try (untested by me):
1) installing an app such as gTasks to sync Google tasks with the mobile device;
2) installing an app such as NitroDesk Touchdown to sync the tasks on the mobile device with the Zimbra server.
If you have Outlook, Thunderbird, Windows Phone, Nokia, etc... you may be able to find software that would enable you to carry-out some other two-step sync.
Be warned it may not be ideal, at least until Google release a tasks API.