The motor has no markings on it, it is installed on a drone (Chinese: Syma X5S make).
I wanted to replace the control board, but unfortunately the cost of import and transportation is expensive. So I thought I could try and do some experiments on it instead.
I suspect the motor is not an ESC based one, as there are only two lines from each of the motors. The control board has an MCU with a watch crystal, and what looks like gyroscope and two N channel dual mosfets for driving the motors.
could you suggest a suitable MOSFET that I can buy online for this application. Also something that works on 3.7V DC would be suitable as I will be powering from a 3.7V battery.
To recommend a suitable MOSFET that wouldn't be an overkill requires to know the current you want to drive.
We know it's a quad so there's 4 motors and it's safe to assume that >90% of the power goes to the motors.
If you tell us the battery capacity and the time it takes for it to discharge (the flight time you get from a fully charged batter) we'll be able to (roughly) calculate the current consumption of each motor and make a wise decision about the MOSFET.