5. In the Cartesian coordinate system, a point whose both horizontal and vertical coordinates are integers is called an "integer point". The image of the quadratic function $y=-x^{2}+6 x-\frac{27}{4}$ and the x-axis enclose a closed figure, which is colored red. The number of integer points inside this red region and on its boundary is ( ).
(A) 5
(B) 6
(C) 7
(D) 8
5.C.

The graph of the quadratic function $y=-x^{2}+6 x-\frac{27}{4}$ intersects the $x$-axis at two points $\left(\frac{3}{2}, 0\right)$ and $\left(\frac{9}{2}, 0\right)$. Between $x=\frac{3}{2}$ and $x=\frac{9}{2}$, there are 3 integers: 2, 3, 4.

When $x=2,4$, $y=\frac{5}{4}$, the integers that satisfy $0 \leqslant y \leqslant \frac{5}{4}$ are 0, 1, so there are 4 integer points;

When $x=3$, $y=\frac{9}{4}$, the integers that satisfy $0 \leqslant y \leqslant \frac{9}{4}$ are 0, 1, 2, so there are 3 integer points.
Therefore, there are a total of 7 integer points.