If you have the Dropbox app installed on your Nook, what you can do is in whatever file manager you are using, long press the file you want to send to Dropbox. A box should come up with different choices (open, edit, delete, etc.). One of those choices should be "share." Select that, and another box will open up with a choice of different programs you have installed that you can share files through - Dropbox should be one of them. Click it, and then you can send the file to your main Dropbox folder or a subfolder you've created. I usually use ES file explorer when manipulating files, but it should work similarly with other file managers.
There are other office apps that allow you to save directly to Dropbox in case you want to switch to one of those (like Quick Office).