How to randomly draw Quaternions within a specific range of Euler angles for rotation?
The simplest way to do this would be to generate the Euler angles randomly, then use them to create the quaternion. Most existing quaternion implementations in game engines and 3D math libraries will have a FromEuler ( Vector3 ) function. 

Since you seem to be rolling your own quaternions rather than using a library, here's the conversion formula.

As a small optimization, since quaternions use half-angles, you could generate random values in the range ±15 degrees (pi/12) and feed these to the sin and cos functions directly.