You can use Date.Month()
for the the month value, here you will get integer value like 1 for Jan...12 for Dec then from this value you can easily get its quarter.
Here is example for you:
Integer iquarter = date.today().month()/3;
if(monthquarter == 0)
system.debug('==FirstQuarter==';
else if(monthquarter == 1)
system.debug('==SecondQuarter==');
else if (monthquarter == 2)
system.debug('==ThirdQuarter==');
else
system.debug('==fourthQuarter==');
Hope this helps!!!
Regards, Krishna