...Have you tried asking directly? Usually words work pretty well.
Caveat: I'm autistic, so I find being direct is the only approach that works.
To me it sounds like they’re a couple but if you must know for sure then the move is to approach the woman when she’s walking the dog alone. Ask her if you can pet her dog, mentioning how you always see her and her boyfriend walking it (and how you always want to pet it). If he’s her brother or anything like that, she’ll correct you.
Although now that I’m rereading your comment it kind of sounds like you’ve already talked to him? I’m not sure but if that’s the case just casually mention how you saw his girlfriend walking the pup the other day or something similar. Just find a natural way to explicitly mention the boy/girlfriend to one of them and see what their response is.
Do they always walk the dog together? If yes, they're probably a couple. If not, sometime when he's walking the dog alone, ask if you can pet the dog. Say the dog's beautiful, what's the dog's name, my name is _____, say you'll see them around. If he's available and interested he'll ask for your number.
