It sounds like what you're wanting to do is to roll your own onboard ESC as part of your quadcopter design to save weight instead of using a prepackaged ESC on its own PCB. That's a perfectly reasonable thing to do, but there's a little more involved than just the microcontroller.
Most importantly, you need a power driver for each motor, either a dedicated BLDC driver IC or 6 discrete MOSFETs. If you use a microcontroller with enough PWM outputs, you won't need a separate BLDC controller IC in addition to the driver. Off the top of my head, there's STM32F0, PIC24, dsPIC, and LPC4350 chips that have dedicated precision motor control PWM timers.
I can't see a datasheet/wiring diagram for the motor you linked, but it has 3 wires (obnoxiously, all black) coming off of it, which means it's likely a standard 3-phase BLDC with no hall effect sensors built in.
The term you want to Google is "sensorless BLDC control", which will come up with a whole bunch of application note PDFs from various microcontroller vendors. This method of control requires use 3 channels (per motor) of your microcontroller's ADC to measure back EMF in the motor in order to tell when to energize the next phase of the motor.