Do I need approval to convert my garage into a room?

Yes

The room is required to be thermally insulated and the conversion work will usually require structural alterations and the installation of services and fittings - therefore approval is required 

If you intend to use the room for any habitable purpose, or erect a new front wall, provide a new door opening, provide a new floor, or install a WC, you are required  to make a Building Regulation application