There's room for a DM to rule both yes and no
The rule for unconscious says:
can't move or speak, and is unaware of its surroundings
It is interesting to note that it does NOT say:
can't move, speak, or hear.
There is no definition of "surroundings" in the game but a plain English understanding would be something along the lines of "all the stuff (physical structures, terrain, creatures, etc) around the unconscious creature.
That "stuff" often produces noise and a character, hearing that noise, would be aware of (some of) her surroundings by definition. In other words, a character can't both be unaware of her surroundings but still hear them. So, given this strict reading of the rules, the Bard can't convey inspiration.
However, the fact that hearing is not literally excluded seems very telling. We know that people can perceive sound when asleep and that even coma patients may still be able to perceive some external stimuli.
So the fact that the rule for unconsciousness is worded a little vaguely suggests that a DM allowing the Bard to convey inspiration would not be out of line given a reading that assumes an intent to simulate actual phenomenon instead of just a strict interpretation of rules wording.