Chinese zodiac is an interesting story, don't believe it more.
In Chinese zodiac, if you older than your girl friend 6 years, it named FanTaiSui.
So, if your animal is dragon and your girlfriend's dog, it means your age older 6 years than her. Yeah, it's not suggesting marry.
But there is a way to solve it if you love each other deeply, you can go temple to pay tribute to TaiSui, then it's nothing

What's TaiSui? and What's FanTaiSui?
Taisui: look at the wikipedia, there is a full infomation about it:
http://en.wikipedia.org/wiki/Tai_SuiFanTaiSui: To give offense to TaiSui, offend him.