Translation:One hour is not shorter than thirty minutes.
Yes, it seems they do. Whether that has always been true is not clear. It may also be that there are Klingon cultures or parts of Klingon society that do not. But there is a distinct implication that at least most, if not all, Klingons will understand time expressed using a 24 hour day.
From Conversational Klingon:
Klingons have adopted the way most civilized planets in the Galaxy tell time. They have twenty-four hour days.
It then goes on to describe normal "military" time. Then it says:
Although Klingons tell time the way most of the rest of the Galaxy does, the Klingon day really goes from dawn to dawn, rather than from midnight to midnight. You might think this could cause some confusion, but it is really comparable to when a Terran says something like Thursday night, at three o'clock in the morning.
This time-telling would be in place on Klingon ships and stations. Naturally, time-keeping on a Klingon planet would depend on the rotation of that planet. Unless its rotation period iwere similar to Earth's, a different number of hours would have to be used. This would be true of Federation planets as well.