It would be hard to establish negligence on the part of the council.
As an analogy, if you trip over a raised flagstone (on a council-owned footpath) you don't have an automatic right to compensation from the council. The flagstone might only have become damaged a few minutes earlier by a lorry driving over it, leaving the council no time to become aware of the problem and arrange for it to be fixed. In order to get compensation you'd have to show that the flagstone had been displaced for some time and that the council either knew about it or, at least, should have known about it (and should had done nothing about it).
So if the branch was only hanging low because of recent winds and rain, it would be unreasonable to expect the council to know about it and thus there was no negligence on the part of the council.
You could try a simple letter to the council, seeking compensation, though (preferably accompanied by photographs of the tree and the damage to the car). It might work, simply because the council might take the view that it wasn't worth the time and hassle of getting into a dispute over such a small sum. (My father tripped over a flagstone, ripping the knee of his trousers. He sent a letter to the council - enclosing his trousers! - and received compensation. However that was half a century ago and councils are now far more likely to stick to the rule of "no proof of negligence = no compensation").