Yes (but not in the way you are thinking)
A blink dog is smart - having the same intelligence, on average, as the average human. They can also speak a language (sylvan).
So if you can train a person you can train a blink dog. But they are not pets - if you want them to do stuff, you will need more than a snack or the promise of walkies - you will need to ...
A blink dog is not a dog.
A blink dog is a fey, not a beast. Furthermore, they understand Sylvan language and are as intelligent, wise and charismatic as the average humanoid, if not more.
You are approaching this from the wrong mindset: a blink dog is a person - you can befriend one, even educate one but you can't train one like it is an animal.
Talk to your DM
The specifics of this situation are really going to be driven by your DM. We can't tell you what to do because this sounds like it's part of the story being told by your table. How you can interact and what the goals of the DM are with the blink dog are really up to them.
I'd recommend just to play as you would, talk to your DM, and see ...