They are shown between quotes because the file name contains backslashes. In the shell, backslashes have a special meaning, so you have to either escape them or quote them.
The quotes are a convenience so you can just copy and paste git status's output, e.g., to add the file to the index:
$ git add "public/images/wallpaper/wait/1920\321\2051080.jpg"
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ copied and pasted
You can just add the filename to .gitignore without the quotes. Most probably, you'll want to add something like this to your .gitignore file:
public/images/wallpaper/wait/*.jpg