Quaternions and translation to Matrix4x4 between two differently define coordinate systems

Unity Engine
1

I have a system providing data in the following coordinate system :
Forward : -x-axis.
Right : -y-axis.
Up: -z-axis.
I am receiving quaternions and translations on those axis.

I want to transform this data to Unity’s coordinate system :
Forward : z-axis.
Right: x-axis.
Up: y-axis.

So there is a switch between the axes and in the directions.

The goal is to put Quaternion Q & Translation T in a Matrix4x4

So, for a quaternion Q I am getting from the system, I write it as :
Quaternion(-y,-z,-x,w) and for a translation T as Vector3(-y,-z,-x)
like in the following code

 Matrix4x4 point = new Matrix4x4();
                point = Matrix4x4.identity;
                point = Matrix4x4.Rotate(new Quaternion(-q.y,-q.z,-q.x,q.w));

                point .SetColumn(3, new Vector4(-ty, -tz, -tx, 1));

Yet I am getting weird movements…
Am I doing anything wrong at this stage ?

2

Z-axis is forward, not back. Matrix columns are coordinate axes x, y and z. Multiply axes by quaternion and fill in the values. After that, fill in the translationin the 4th column. And put 1 into m33. If you wish reverse transformation inverse the matrix.