Il est quinze heures et quart.

Kama, 

it is just not said that way in France when using the 24 hour clock. 

Neither would it normally be said that way in English when using the 24 hour clock, as it is about precision and removing as much risk of time confusion as possible. Using ‘ quart ‘ or ‘ quarter ‘, is not considered " precise/safe ". Neither is ‘ et demie ‘ or ‘ half past ‘ combined with 24 hour times in the respective languages. In both languages ‘ 24 hour clock ‘ times use exact numbers for the hour, and the minutes, ( and the seconds if needed).

The lesson covers it : 

When the time expressed uses hour numbers above 12 (in the "24-hour clock" -> 13h, 14h...), you use instead quinze, trente, quarante-cinq, probably for pronunciation (and elegance) reasons. 

The attached link may be useful for reinforcement of the lessons here.

https://www.frenchtoday.com/blog/french-vocabulary/how-to-tell-the-time-in-french/

Is the converse also true, ie the 24 hour clock isn’t used in French (orally) for times from midnight to midday? In English you can say oh-three-fifteen to show precisely that you mean 03:15hr.