Can Truss Rod adjustment lead to String break?
When you tighten the truss rod, it increases to a degree the tension on the strings. For a small adjustment, that slight difference should not cause a healthy string to break. But if the string was already compromised, the slight increase in tension might be enough to stress it to the breaking point.
Even without any increase in tension, loosening and then tightening a weak string could be enough to snap it, because the precise dynamics of the tension on the string have changed slightly: Such changes could exacerbate/expose the weakness of the string and cause it to break.
Did i mess up my Guitar?
You can damage your guitar by tightening the truss rod excessively. You can damage the truss-rod, and untoward stress on the neck can cause neck buckling, fret buckling, weaken its structural integrity etc - all sorts of bad things, many of which will only be manifest over a period time. But it's unlikely that a quarter turn would cause that, unless you were already at "the breaking point" before you started. So you needn't to worry about that, assuming it was new when you got it, unless you've tightened it a quarter turn not just this once when the string broke, but a few times.
(Just keep in mind that if one morning you find your guitar with the neck cracked open for no apparent reason, it might be because you overdid it with the truss rod.... just joking - sort of :) )
As @Tim mentioned in the comments, whenever you adjust the truss rod or the action, go slowly and incrementally, and make you sure you know how to get back to your starting point. Here are two good links about the subject:
Ask the Expert: How Does a Truss Rod Work?
Guitar 201: Adjusting the Truss Rod (Very detailed)
When in doubt bring your guitar to a good guitar tech, to do a set up for you. It's the cheapest, best way to improve your guitar: Many problems that are often blamed on the quality or manufacture of the guitar are actually the results of poor set-ups. (Factory set-ups are often far from good, same when you buy something off the rack in a store.)
As it stands, you should be fine just putting on a new string (Or maybe a new set of matched string that are all at the same starting point) .
Go play your guitar - enjoy yourself.