compute the discrete(sampled) time process noise matrix Given continuous time state transition equation as follows, 
$$
\frac{dx}{dt}=Ax+\nu
$$
where $\nu \sim N(\mathbf{0},Q)$, $Q$ being the process noise matrix, one can compute the discrete(sampled) time process noise matrix as follow,
$$
Q(\delta t)=\int\limits_{0}^{\delta t}{{{e}^{A\tau }}Q{{e}^{{{A}^{T}}\tau }}d\tau }
$$
I want to ask why.
 A: Indeed this is used often in electrical engineering such as design of Kalman filters, if I recall correctly.
$$\mathbf Q(k)=\int\limits_{k\,T}^{(k+1)\,T}{{\mathbf \phi((k+1)T,\tau)\;}\mathbf Q(\tau){\;\mathbf \phi((k-1)T,\tau)}^\top d\tau }$$
gives the definition of the discrete-time noise covariance (usually defined for a certain time interval) it follows the mathematical definition of covariance and autocorrelation. Herein $\mathbf Q(t)$ is the continuous time process noise covariance matrix. The integral can be also approximated by a sum.
In your notation, $\delta t$ corresponds to the piece of time interval $[kT,(k+1)T]$. Further, $\phi$ shall represent the variable of state or, better said, through your first equation determine the state transition. In your case this is the solution of your first differential equation (noise excluded of course) detemining state transitions following $\phi(t)\sim x(t)=\exp(\mathbf A t)$, hence:
$$\mathbf \phi(\tau)=\exp(\mathbf A \tau)$$
$$\mathbf \phi^\top(\tau)=\exp(\mathbf A^\top \tau)$$
where ${}^\top$ indicates transposition, all capital bold symbols are matrices and $\phi$ is a vector.
Once you apply the above integral to your case, you will see that the matrix $\mathbf Q$ acts with its elements operating as autocorrelation of the continuous process noise on the variable of state — these are commonly denoted $\sigma_{ij}$ — where $i,j$ are some indices.
