Quarter-hour would be your best bet, although native speakers tend not to use this in regular conversation. Quarter of an hour is also correct, but, I daresay, less likely to be used. The phrase, every fifteen minutes can often be heard in everyday usage.
For the specific software situation you mentioned, again, quarter-hour would be your best bet. On its own, quarter in temporal contexts usually refers to quarter of a year.
The armed services (and their veterans) really have this engrained in my mind as such:
Rendezvous at 0600 [O-Six-Hundred] hours!
Drop point is X degrees north, at 1200 [Twelve-Hundred] hours!
Otherwise, where more precise in terms of declaring minutes, you can just split them and speak each unit of time individually:
Your meeting is at 1530 [Fifteen-Thirty] hours.
Your meeting is at 1812 [Eighteen-Twelve] hours.
The hours here, it might be argued, is redundant or even inaccurate, but that doesn't dictate the occurrences (or exclusion of such) in speech.
You could go the quarter to, half past route, but this is an interchangeable method of speaking time, not exclusive to verbalising time in its 24-hours form.
Since o'clock is an abbreviation of of the clock, I guess that technically you could speak in this manner in terms of 24-hours, such as: 15 o'clock. But this might sound a little peculiar to most. If we look at the definition of o'clock from TheFreeDictionary then it kind of indicates we would be playing it safer to use another form:
Of or according to the clock: three o'clock.
According to an imaginary clock dial with the observer at the
center and 12 o'clock considered as straight ahead in horizontal
position or straight up in vertical position. Used to indicate
relative position: enemy planes at 10 o'clock.
used after a number from one to twelve to indicate the hour of the
day or night
(Mathematics & Measurements / Navigation) used after a number to
indicate direction or position relative to the observer, twelve
o'clock being directly ahead or overhead and other positions being
obtained by comparisons with a clock face
If we do decide to use the 24 o'clock approach, then it's just redundant, if nothing else; consider the note on relativity to the face / direction. Since, regardless of the numbers being bigger, we don't have to (necessarily) do any extra laps around the clock face to arrive at the specified location - but in cases where AM and PM might not be clearly implied, it could serve to do that.
Best Answer
From wikipedia:
From a military forum: