Can I miss any of the "ambient conversation" quests?

Solution 1:

If the dialog started, and that dialog is supposed to give you a Journal entry, it will even if you don't stick around to let it finish.

I had this happen in the entrance to the Purgatory Bar in the citadel -- I listened to a woman talking on the phone (all the conversation) and got a mission, then I had to go and exited the game forgetting to save (silly me) and had to replay that part. I didn't feel like listening to it all again so I just kept running and a few moments later I got the same mission even though I was obviously out of earshot.

So just be sure to run passed all the people you see, but don't feel obligated to listen to them (although most conversations are really entertaining)

Solution 2:

I'm not sure if you can miss getting them, but you certainly can miss completing them

When Cerberus attacks the Citadel some of these people are killed in the crossfire.