I bought the G.O.T.Y edition of Fallout 3 and it's working just fine. The only problem I have is that I haven't gotten any quests for the dlcs. I don't know why this is but any help would do.

In the first start up splash screen go to Data Files and make sure everything is checked.

I have them installed but they're not showing in the data files thingy.

Try to use a mod manager, and tick the DLCs there (if they are not already)

This is trickier to know what may be happening or how to fix.
Have you had other releases installed (from steam or original DVD media) then installed the GoG release?
First, confirm the DLC files are in your Fallout3 game install folder in the Data folder.
You can also check for fallout.ini files in the main Fallout3 install folder, remove them.
Then check "Documents\My Games\Fallout3 and remove those ini files (you should be able to keep the Saves).
Restart Fallout, reconfigure video settings, check Data Files section for everything to be checked.

If you use mods, and a mod manager (vortex) it is possible it did something. You can check in there and see if it sees the DLC under... uh... forget what it is called on left side, but there is an area that shows just the bsa/esp type files that would show up in the Data Files area in the launcher UI. Mods can complicate things further so that is about all I will go into here.

I know I'm a few years late, but for anyone in the future who happens to stumble upon this.
What worked in my case was going through the games files, opening "GECK", select "File" then "Data",
then double click each DLC you want to enable, make sure they are labelled with a cross in the box next
to the name of the DLC and then press "ok", another window should pop up, just click "ok" again. close
GECK. To confirm they are loaded in, open the launcher and select "Data Files" and they should be toggleable now,
while I can't say much for this last step in how it helps if at all, click "Load Loose Files" and click "ok".
Try booting the game and soon it should start loading in the DLC.

Hope this was helpful.