In the movie Skyfall Shouldn't Q like connect Silva's computer to a secure line or something, considering that he was a known cyberterrorist? Isn't there any sort of internal protocols for the networking system?
There is no proper reason whatsoever for him to link the unsecure device retrieved from an enemy to the network of MI6. The usual approach would be to treat it with the highest level of suspicion and analyze the hardware and contained data in a secure isolated environment, and Q should have known that, considering that he is supposed to be a genius IT specialist.
This is a trope called the Idiot Ball (warning: TvTropes link!):
If you are looking for a reasonable in-world explanation, one could come up with the following reasons for his mistake:
Still, that was a very stupid thing to do and for me was one of the biggest facepalm moments in the movie.
While I don't remember that specific scene (not sure I've even seen the movie), this sounds once again quite like something the writers made up just for the story. As such it's rather hard to answer why Q would do it (because it's the same reason: just for the plot).
Unless you've got some really nasty bug in your operating system or some insecure autorun policies set, there shouldn't happen anything once you connect a drive (and you don't try to execute something on it). Everything on it is just inactive/passive data.
There could be additional safeguards built into the hardware (similar to drive encryption), but once again that's unlikely to cause any side-effects outside; at least nothing software only (other than potentially reading wrong information).
If he intended to execute something on the drive, then yes, you're right, it's very careless.
Although, to be honest, I'd try to create a copy of the drive (so it's unable to destroy the data on it itself) and then mount that in a virtual machine (so it won't be able to destroy my soft-/hardware). It's not just about disconnecting from the network (you'd most likely do it on isolated hardware to begin with; so you wouldn't have to worry about disconnecting first).