It would be correct to say "我们的狗有绿色的眼睛。"
I guess, grammatically, it would also be correct to say "绿的眼睛", but it does not sound so good to my ears in this sentence. Either say "绿色的眼睛" or "绿眼睛", that sounds good.
We're just lucky they didn't decide to be creepy this time. "Our dog has a pair of green eyes, and a pair of brown eyes, and four or five blue eyes, all buried around the yard. We're starting to worry where he's getting them from."