implementing right handed coord system APIs on top of a left-handed rendering engine
What you probably need is this 4 x 4 matrix:

$$\mathtt{T} = \begin{bmatrix}1 &amp; 0 &amp; 0 &amp; 0\\ 0 &amp; 1 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; -1 &amp; 0 \\ 0 &amp; 0 &amp; 0 &amp; 1 \\  \end{bmatrix}$$
which is flipping the z-axis. The inverse of this particular matrix is the matrix itself:
$\mathtt{T}^{-1} = \mathtt{T}$

For 3D points $\mathbf{x}$ multiply it from the left:
$$\mathbf{x}' = \mathtt{T}\,\, \mathbf{x}$$
and for 3D transformations multiply it from the right:
$$\mathtt{P}' = \mathtt{P} \,\,\mathtt{T}$$

Thus, you have not changed anything in the projection because:
$$\mathtt{P}'\,\,\mathbf{x}' = \mathtt{P} \,\,\mathtt{T} \, \,\mathtt{T}\,\,  \mathbf{x} =  \mathtt{P} \,\,(\mathtt{T}^{-1} \, \,\mathtt{T})\,\,  \mathbf{x} = \mathtt{P}\,\,\mathbf{x}$$

Quaternions $q=(s,x,y,z)^\top$ could be convert into 
a 3D transformation matrix with:

$$\mathtt{P} = \begin{bmatrix} 1-2y^2-2z^2 &amp; 2xy-2sz&amp; 2xz+2ys &amp; 0\\ 
              2xy+2sz &amp; 1-2x^2-2z^2&amp; 2yz-2sx &amp; 0\\ 
              2xz-2sy &amp; 2yz+2sx&amp; 1-2x^2-2y^2 &amp; 0\\  0 &amp; 0 &amp; 0 &amp; 1\end{bmatrix}$$
and could then be treated like other 3D transformations.

However, 3D engines typically apply transformation in a transformation hierarchy. If this is the case, make sure you are only modifying the 3D transformation "closest" to the 3D point in the hierarchy.

Sources:
Graphics Programming I - 3D Transformations | Thorsten Thormählen