The magic buttons represent the malleability of physical reality and also the openness of any world, whether real or fantasy. The buttons symbolize the fact that physical objects can be repurposed to be used for whatever reason their creator has in mind. The buttons demonstrate that magic is a craft, and that it uses physical reality as a material to sustain or enhance the world of magic. In this case, the buttons are used as a teleportation device to get in and out of the Neitherlands/Fillory. However, the purpose of their use also points to the fact that they can breach the rules by which magic and fantasy define their worlds. Under the law in Fillory, outsiders are allowed in only by invitation and only if they’re worthy. The magic buttons can bypass those rules, and enable anyone to come to Fillory. This implies their origin came from something or someone other than those who created and govern Fillory. 

For Quentin, Brooklyn is a symbol of the banality of the real world. Brooklyn is the dreariness of his parent’s home, the melancholy weight Quentin sees in Julia’s eyes, and the numbness that leads Quentin and Eliot to seek refuge in alcohol.