I was sent by the caption of the guard in Vivec Hall of Justice to find a skirted lady in a netch leather cuiress and armed with a dagger. After asking around, I found her in the underworks of the Foreign Quarter and was immediately attacked. I was naturally forced to kill her in self defense. The game identified her as the "dreamer women". When I returned to the Captain, he did not seem to know that the quest had been completed, and there was no way to tell him so. I checked my journal, and there was no indication there that I had completed the quest. What went wrong here? Was this the lady I was supposed to find?

In the underworks of another canton (can't remember which) I also saw a man the game identifed as 'dreamer'. He did not attack as the women did, so I left him alone. Is this man related to this quest?

In answer to your two questions:-

There are several places where dreamers appear, more later in the game. They do not usually attack (occasionally one attacks you while you are asleep) and are not related to a quest. They are really background colour for the main quest.

The woman with the dagger has to be killed and doing so should get you a journal update. The first thing to check is does the quest actually appear in the open quest section of your journal. Sometimes you can think you have agreed to do a quest but in fact you have omitted a particular connection. (The fighters guild quest re Lirielle Stoine's debt is a good example of this.) If the quest is in your journal it may be that another connection is needed. You can check that out in the CS. I haven't got the game at work so don't know but maybe you need to talk to someone in the foreign quarter to learn about the dunmer woman with a dagger before finding her.

A third possibility is you are reporting back to the wrong man but I suspect that is unlikely.

If the worst comes to the worst you can update the journal via the console.