Google Task API Move Task to different list

The tasks API does not support moving tasks from one list to another, just changing a tasks parent task or moving it to the top of the list.

You can however get the task with /lists/tasklist_id/tasks/task_id, check each of its fields and set populate the same fields with a different tasklist_id using /lists/new_tasklist_id/tasks and then delete the old one with /lists/tasklist_id/tasks/task_id

Google also provides the Tasks API explorer which should help you experiment.

Just to confirm the answer by Thai Wood:

This is definitely the way to do it. I also moved a task using the website and and noticed that the task id changes, therefore the google website also seems to do a delete followed by a insert.