Let $B = \left({A\atop -A}\right)$ be a matrix, whose rows are formed by the rows of matrices $A$ and $-A$. Then $$C = \{ x\in\mathbb{R}^d \mid Bx \leq u \},$$ where $u=(\underbrace{1,1,\dots,1}_{2d})^T$, is a convex polyhedron. Furthermore, $$Q_d \cap P_d = \{ Ax \mid x\in C\cap \mathbb{Z}^d \}$$ and thus the problem is reduced to enumerating the integral points in the polyhedron $C$.

There is a readily available software called LattE for lattice point enumeration, which can find the integral points of $C$ in the form of multivariate generating function (with the command "count --multivariate-generating-function"). While it requires elements of the matrix be integer, it is not hard to achieve that by taking an appropriate rational approximations and multiplying $Bx\leq u$ by an integer constant to make all entries integer.

See LattE documentation for further references and examples.

Let $B = \left({A\atop -A}\right)$ be a matrix, whose rows are formed by the rows of matrices $A$ and $-A$. Then $$C = \{ x\in\mathbb{R}^d \mid Bx \leq u \},$$ where $u=(\underbrace{1,1,\dots,1}_{2d})^T$, is a convex polyhedron. Furthermore, $$Q_d \cap P_d = \{ Ax \mid x\in C\cap \mathbb{Z}^d \}$$ and thus the problem is reduced to enumerating the integral points in the polyhedron $C$.

There is a readily available software called LattE for lattice point enumeration, which can find the integral points of $C$ in the form of multivariate generating function (with the command "count --multivariate-generating-function"). While it requires elements of the matrix be integer, it is not hard to achieve that by taking an appropriate rational approximations and multiplying $Bx\leq u$ by an integer constant to make all entries integer.

See LattE documentation for further references and examples.

Let $B = \left({A\atop -A}\right)$ be a matrix whose rows formed the rows of matrices $A$ and $-A$. Then $$C = \{ x\in\mathbb{R}^d \mid Bx \leq u \},$$ where $u=(\underbrace{1,1,\dots,1}_{2d})^T$, is a convex polyhedron. Furthermore, $$Q_d \cap P_d = \{ Ax \mid x\in C\cap \mathbb{Z}^d \}$$ and thus the problem is reduced to enumerating the integral points in the polyhedron $C$.

There is a readily available software called LattE for lattice point enumeration, which can find the integral points of $C$ in the form of multivariate generating function (with the command "count --multivariate-generating-function"). While it requires elements of the matrix be integer, it is not hard to achieve that by taking an appropriate rational approximations and multiplying $Bx\leq u$ by an integer constant to make all entries integer.

See LattE documentation for further references and examples.

Let $B = \left({A\atop -A}\right)$ be a matrix, whose rows are formed by the rows of matrices $A$ and $-A$. Then $$C = \{ x\in\mathbb{R}^d \mid Bx \leq u \},$$ where $u=(\underbrace{1,1,\dots,1}_{2d})^T$, is a convex polyhedron. Furthermore, $$Q_d \cap P_d = \{ Ax \mid x\in C\cap \mathbb{Z}^d \}$$ and thus the problem is reduced to enumerating the integral points in the polyhedron $C$.

There is a readily available software called LattE for lattice point enumeration, which can find the integral points of $C$ in the form of multivariate generating function (with the command "count --multivariate-generating-function"). While it requires elements of the matrix be integer, it is not hard to achieve that by taking an appropriate rational approximations and multiplying $Bx\leq u$ by an integer constant to make all entries integer.

See LattE documentation for further references and examples.