In theory a leap second can happen at the end of a quarter. However since their introduction they have only ever happened at the end of a half i.e. 30th of June or 31st of December.
On these dates when a leap second event occurs the UTC time sequence would be:
23:59:58
23:59:59
23:59:60 (sometimes 23:59:59 is written again)
00:00:00