The quaternion representation of rotation is a variation on axis and angle. So if you rotate by r radians around axis x, y, z, then your quaternion q is:
q[0] = cos(r/2);
q[1] = sin(r/2)*x;
q[2] = sin(r/2)*y;
q[3] = sin(r/2)*z;
If you want to create a quaternion that only rotates around the y axis, you zero out the x and z axes and then re-normalize the quaternion:
q[1] = 0;
q[3] = 0;
double mag = sqrt(q[0]*q[0] + q[2]*q[2]);
q[0] /= mag;
q[2] /= mag;
If you want the resulting angle:
double ang = 2*acos(q[0]);
This assumes that the quaternion representation is stored: w,x,y,z. If both q[0] and q[2] are zero, or close to it, the resulting quaternion should just be {1,0,0,0}.