Can you have 12 weeks in a payroll quarter and 14 weeks in a different payroll quarter for the same year and the other quarters are all 13 weeks?

Yes, that happens in regular years (non-leap years) that start on the weekday after the usual pay weekday. In such a year, the first quarter has 12 paydays and the third quarter has 14 paydays. So since 2013 starts on a Tuesday, 2013 will be a year like that for people who are paid every Monday.

A quarter is about 13 weeks. Every 2nd quarter and the 1st quarter of every leap year is exactly 13 weeks. Therefore, those quarters will always have 13 weekly paydays, no matter what weekday is the first day of the quarter. The 1st quarter of every regular year is one day short of 13 weeks, so if Jan. 1 is the weekday after the usual pay weekday, that 1st quarter has 12 paydays. If Jan. 1 is any other weekday, that 1st quarter has 13 paydays. Every 3rd and 4th quarter has 13 weeks plus one day. So whenever the first day of a 3rd or 4th quarter is payday, the last day of the quarter is, too, giving you 14 paydays. Every time a 1st quarter has 12 paydays, the first day of the 2nd and 3rd quarters will be payday. So every time you lose a payday in the 1st quarter, you get it back in the 3rd quarter.