Example 1 Find $a, b, c$, such that they satisfy the inequality $a^{2}+b^{2}+c^{2}+3<a b+3 b+2 c(a, b, c \in \mathbf{Z})$.
Given that integers $a, b, c$ satisfy $a^{2}+b^{2}+c^{2}+3<a b+3 b+2 c$,
i.e., $a^{2}+b^{2}+c^{2}+4 \leqslant a b+3 b+2 c$.
Completing the square, we get $\left(a-\frac{b}{2}\right)^{2}+3\left(\frac{b}{2}-1\right)^{2}+(c-1)^{2} \leqslant 0$.
Since $\left(a-\frac{b}{2}\right)^{2} \geqslant 0,\left(\frac{b}{2}-1\right)^{2} \geqslant 0,(c-1)^{2} \geqslant 0$,
it follows that $a-\frac{b}{2}=\frac{b}{2}-1=c-1=0$,
i.e., $a=1, b=2, c=1$.