Quaternion is another representation of axis angle. The solution is to create a new quaternion from the original quaternion that only has the needed components.

An axis angle representation can be converted to a quaternion using the following formula

q[0] = cos(R/2);
q[1] = sin(R/2)*x;
q[2] = sin(R/2)*y;
q[3] = sin(R/2)*z;

Where R is the angle in radians, and (x,y,z) represents the axis, and quaternion is (R,x,y,z).

So in order to create a new quaternion with only the pitch component you just zero out the other components and normalize the quaternion:

Quaternion q; // this is your original quaternion
q.x = 0.0;
q.z = 0.0;
q.Normalize();

And we are done.

Quaternion is another representation of axis angle. The solution is to create a new quaternion from the original quaternion that only has the needed components.

An axis angle representation can be converted to a quaternion using the following formula

q[0] = cos(R/2);
q[1] = sin(R/2)*x;
q[2] = sin(R/2)*y;
q[3] = sin(R/2)*z;

Where R is the angle in radians, and (x,y,z) represents the axis, and quaternion is (R,x,y,z).

So in order to create a new quaternion with only the pitch component you just zero out the other components and normalize the quaternion:

Quaternion q; // this is your original quaternion
q.x = 0.0;
q.z = 0.0;
q.Normalize();

Edit based on your update:

Sensors AFAIK calculate orientation relative to gravity, in other words Y (or Z) is the direction of the gravity. You need take that into consideration. And I think you don't need to multiply it with the inverse initial orientation.