Update – 15/10/2012
Thanks to Marcin, I discovered this addon to Lightning on Thunderbird. Basically it’s an addon to Lightning’s “Today Pane”, where you can now view and and edit your tasks on Google! It’s still not the complete integration you could have with tasks, so please continue to follow the instructions below.
[Vote for this issue and get email change notifications] Your vote has been recorded.
No, this is not possible at this time with Lightning and the Google Calendar Provider. If you want to have a chance of getting this working in the near future, without working on the code yourself, then vote for this issue.
To vote, click on the link above and head to the bottom of the page where it says, “Add a star” and click on the star next to it. Do this, and ask everyone you know, who uses Thunderbird with Gmail, to do the same. Hopefully, our votes will bring this much needed feature to fruition.