Count values in range. Possible?

http://i.imgur.com/EEqMDUa.png

We have a start date and an end date for a project. We need to figure out the number of days that fall into each quarter so billing can be done appropriately.

The basic logic is that I have the start date and end date of each quarter, and would like to count the number of values that fall into each range. I can't find a "count between" function. Any help is appreciated!