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Nostalgia is entirely a subjective experience. Even if your dog would tell you that it does you'd still have to take their word for it.
Other human beings are the same problem, no one can tell how others really feel. Let's get solipsism out of the way, so that we can talk practically.
Human beings share a lot of genetics and brain functions - we can study that. And they share a lot of genetics and brain functions with dogs too! I don't think nostalgia in dogs has been studied yet, but I do think we can get an answer one day.
I mean, there's a good chance dogs can experience nostalgia. I'd say that it's even likely. However, my point is that we can't know for sure and I'd argue that we may not ever find out. Nostalgia is a subjective experience that appears in consciousness and just like consciousness itself, there zero evidence of it in the world outside of your own experience of it. How would you study something that can't even be detected?
I'd study it in humans first - the advantage is they can talk. I'd study the way brain works in nostalgia feeling humans. Then I'd try to find the same or similar functionality in dog brains (which are fortunately similar enough, because dogs are evolutionarily relatively close to humans).
Let me again state that feelings are subjective even in humans and you could argue that what others really experience can be different even though it manifests the same externally. I'm knowingly ignoring this phylosofical side of the problem and I'm proposing a practical scientific way to approach it.