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 where the story goes.
Can fey be trained?
I don't know the answer to this as the rules don't cover this. When the rules don't cover something, it is up to the DM to determine. Ruling either way is acceptable (after all, humans are animals and we're clearly trainable) and it will come down to how the table and the DM wants to treat it.
Maybe trying to train it is part of the story your DM is telling and this is a character growth opportunity. Who knows? Your DM.
But, either way, you need to establish trust with this fey. The blink dog may have been attracted to the toy from its master because it was it's favorite toy AND because it trusted its master.
You've only got half the equation: the toy. Now you need to work on building trust and a relationship.