The points $P$ and $Q$ are the midpoints of the edges $AE$ and $CG$ on the cube $ABCD.EFGH$ respectively. If the length of the cube edges is $1$ unit, determine the area of the quadrilateral $DPFQ$ .
1. **Identify the coordinates of the points:**
   - Let the cube be positioned in a 3D coordinate system with vertices:
     \[
     A(0,0,0), B(1,0,0), C(1,1,0), D(0,1,0), E(0,0,1), F(1,0,1), G(1,1,1), H(0,1,1)
     \]
   - The midpoints \(P\) and \(Q\) are:
     \[
     P = \left(\frac{0+0}{2}, \frac{0+0}{2}, \frac{0+1}{2}\right) = (0,0,\frac{1}{2})
     \]
     \[
     Q = \left(\frac{1+1}{2}, \frac{1+1}{2}, \frac{0+1}{2}\right) = (1,1,\frac{1}{2})
     \]

2. **Determine the coordinates of \(D\) and \(F\):**
   - \(D(0,1,0)\)
   - \(F(1,0,1)\)

3. **Calculate the lengths of the diagonals \(DF\) and \(PQ\):**
   - For \(DF\):
     \[
     DF = \sqrt{(1-0)^2 + (0-1)^2 + (1-0)^2} = \sqrt{1 + 1 + 1} = \sqrt{3}
     \]
   - For \(PQ\):
     \[
     PQ = \sqrt{(1-0)^2 + (1-0)^2 + \left(\frac{1}{2} - \frac{1}{2}\right)^2} = \sqrt{1 + 1 + 0} = \sqrt{2}
     \]

4. **Calculate the area of the quadrilateral \(DPFQ\):**
   - Since \(DPFQ\) is a rhombus, the area can be calculated using the formula for the area of a rhombus:
     \[
     \text{Area} = \frac{1}{2} \times \text{diagonal}_1 \times \text{diagonal}_2
     \]
     \[
     \text{Area} = \frac{1}{2} \times \sqrt{3} \times \sqrt{2} = \frac{1}{2} \times \sqrt{6} = \frac{\sqrt{6}}{2}
     \]

The final answer is \(\boxed{\frac{\sqrt{6}}{2}}\).