Which types of apples are considered to be "quick-cooking" apples?
One of my favorite topics, having grown up close to two apple orchards...

Most likely, by "quick-cooking," the recipe intends you to use a pie or sauce apple, i.e. one that softens readily with heat.


Sauce apples. Use these for a pie if you like VERY soft pie contents. Personally, I prefer applesauce that has some chunks in it, so I don't use "sauce apples" for sauce; however, the standard is to list for sauce those apples that practically dissolve (like McIntosh). 
Pie apples. An apple listed for pie is typically one that retains its shape but softens well (like Cortland, Mutsu, Empire, Jonagold, or Fuji).
Consider the taste. Some apples (Gala, for example) lose a lot of flavor when cooked, and are best for eating raw. Others gain tremendous flavor when cooked (Empire). 


Which apple to use is certainly a matter of preference. Some people like their pie apples to remain quite firm (using, say, Granny Smith), while others like them to be VERY soft (and thus use a "sauce" apple).

Here's some apple lists/charts ... I'd say look up the varieties readily available to you, and see which ones are listed for sauces or pies.


http://www.baumanorchards.com/chart/index.htm
http://www.recipegoldmine.com/kitchart/kitchart71.html


Look up the website for your local orchard - they may link to a usage chart for the apples they grow!