The things that are common to the position of Short Stop and QB, to name a few: Accuracy (not saying that all SS or QBs have this but all good ones do), mobility (again, not saying they all have it but...), the ability to throw from different platforms (ditto). Now, of course all players and situations are unique so it stands to reason that there is no universal statement of fact in any of this but, these are the things that you desire in both positions and things that make players at both positions desirable.
The huge differences IMO are, size and throwing mechanics for each sport. As a rule, a lot of SS are not overly big. QBs are typically bigger and it matters in the NFL. Lots of QBs are or were former Baseball players and That's one of those deals where you understand the differences in throwing motions etc. However, if you try and take a SS and turn him into a QB, it doesn't always translate because it's a completely different mechanic to throw a football, as opposed to a baseball. You might be able to be accurate and all the rest with a Baseball but that doesn't guarantee that you will be accurate with a pig skin. A perfect example of this is Henson, a few years back. If you watch his throwing motion in College, it was much better then when he came back from the Yankees and joined the Cowboys. At Michigan, he was a can't miss QB prospect. When he came back, his throwing motion was that of a Third Baseman.