Pure Java-8+ solution:
LocalDate localDate = LocalDate.now();
LocalDate firstDayOfQuarter = localDate.with(localDate.getMonth().firstMonthOfQuarter())
.with(TemporalAdjusters.firstDayOfMonth());
LocalDate lastDayOfQuarter = firstDayOfQuarter.plusMonths(2)
.with(TemporalAdjusters.lastDayOfMonth());