You can do it many ways applying your creative ideas. You can set the Starting Slide Image as a Birthday Image. The image will be linked to a Page or Post where you will describe your history, birthday events, any special promotions etc.
You can also set the Special Text for Featured Content Birthday Deal or something like that.
You may find any appropriate Plugin at Plugins > Add New searching with appropriate keyword
You can set something related Birthday with your logo image like Google Doodle.