Using GPS based navigation alone is unlikely to yield the precision and control rate sufficient for navigating a quadricopter. This is not merely due to limitations in the quality or precision of GPS modules, but the inherent and by-design low precision data as available on civilian GPS signals from the satellites.
Besides, if the quadricopter were to lose "visibility" to the satellites for any period, such as when flying under trees, inside buildings, or under heavily overcast clouds, the navigation system goes for a toss.
What you need to be considering is an Inertial Measurement Unit (IMU) module, perhaps one with "11 Degrees of Freedom". While the DOF term is largely misused marketing buzz, the so-called 11-DOF IMUs typically will include a GPS, as well as magnetic compass and gyroscope for navigation, and perhaps also a barometer for altitude sensing. Prices are not very friendly though.
In order to keep costs down, the alternative is to use a basic (6-DOF) IMU and a separate GPS module, both connected to your Arduino, with code to process and utilize this sensor data for navigation. A quick web search would yield many blogs and tutorials to help you put this all together.
Sites like SparkFun, SeeedStudio and AdaFruit offer IMU and GPS modules that work with Arduino.
Some additional research is suggested before actually starting to order the parts.