4x4 transform matrices typically represent 3-space transformations with homogeneous coordinates. Rotation matrices in this framework are effectively the same thing as Euler angles. They are not quaternions.
If you want to get pedantic they're also not euler angles, they're 4x4 transformation matrices.
You can pick out euler angles just as easily as calculating the quaternion by taking the basis vectors through the matrix and extracting it from there.
You can pick out euler angles just as easily as calculating the quaternion by taking the basis vectors through the matrix and extracting it from there.