Yes, it can be done (in a manner of speaking), because I've done it myself.
First, make your original mission, test it, save it, and publish it.
Next, change the name of your original mission to something else and save it again.
Next, go and Import your original mission from the server.
You will now have two identical missions with different names and, if everything has worked right, different hashes. That is the important part. As long as the hashes are different then they are considered different missions to the server.
Now take your original mission (with the new name) and make whatever changes you want to transform it into your new one. Once done, publish the second mission as normal.
While this does not create one mission with different goals per se, it does allow you to make two seperate yet similar templates. This is how I did "Repo Man Blues" - the original is level 41 only because of the Borg in it. There is a second version open to all levels because I replaced the Borg with Orions.
There's no easy built in way of making branching missions yet in the Foundry, but to get something that feels that way, watch the tutorial videos on starbaseugc.com on using triggers and branching dialogues.
Here's a setup you can do using the Foundry as it is now.
You have a map, and you place things on the map that move along the story. You plan to put enemies on the map, but storywise it doesn't matter what kind of enemies they are.
At the start of the map, you can ask the player - do you want Borg or Breen?
Depending on which option they choose, you can then use a trigger to populate the map with either Borg that beam in or Breen that beam in.
The limitations are that:-
1) the enemy groups must be optional - you can not set objectives to kill them because the game won't know which ones you are beaming into the map - Borg or Breen?
2) Your choice is limited to the map you are currently on. The game does not remember choices a player makes when they switch to a new map.
Another way you could do that is have a map and split it into two halves - at the start of the map there's a door that leads to a bunch of Borg and another door that leads to a bunch of Breen. Depending on what the player chooses they can go through a door and fight what they want. The objective would be to reach the other side of the map.