Let $X$ be the blow up of $\mathbb{P}^{2}$ in a torus invariant point. The blow up is toric, in fact its moment polytope can be calculated by "cutting away" a triangular region around the vertex corresponding to the fixed point (texts on toric geometry will cover this, for example the book of Cox).

As with all blow ups (in dimension $2$), the exceptional divisor $E$ is isomorphic to $\mathbb{P}^{1}$ with self intersection number $E \cdot E = -1$. For a complex algebraic manifold $X$ we have that the first chern class of the tangent bundle is Poincaré dual to the anticanonical divisor $-K_{X}$. Now lets apply the adjunction formula to $E$.

$$K_{E} = (K_{X} + E)|_{E} $$

We know $$\int_{E} K_{E} = -2, $$ since it is topologically a $2$-sphere. Hence $$\int_{E} c_{1}(X) = \int_{E} -K_{X}|_{E} = \int_{E}(-K_{E} + E)_{E} = 2 -1 = 1.$$

Hence the minimal Chern number is $1$. Given any toric surface we can blow up a torus invariant point and the same calculation will give that the minimal chern number is $1$.

So the only remaining cases to check in dimension 2 are minimal models. Since they are rational there is only $\mathbb{P}^{2}$ and the Hirzebruch surfaces $\mathbb{F}_{i}$. The minimal Chern number of $\mathbb{P}^{2}$ is 3, the minimal Chern number of $\mathbb{P}^{1} \times \mathbb{P}^{1} = \mathbb{F}_{0}$ is 2. The minimal Chern number of all the other Hirzebruch surfaces is $1$. Hence there are only two (smooth projective) toric surfaces with minimal Chern number not equal to $1$.

Let $X$ be the blow up of $\mathbb{P}^{2}$ in a torus invariant point. The blow up is toric, in fact its moment polytope can be calculated by "cutting away" a triangular region around the vertex corresponding to the fixed point (texts on toric geometry will cover this, for example the book of Cox).

As with all blow ups (in dimension $2$), the exceptional divisor $E$ is isomorphic to $\mathbb{P}^{1}$ with self intersection number $E \cdot E = -1$. For a complex algebraic manifold $X$ we have that the first chern class of the tangent bundle is Poincaré dual to the anticanonical divisor $-K_{X}$. Now lets apply the adjunction formula to $E$.

$$K_{E} = (K_{X} + E)|_{E} $$

We know $$\int_{E} K_{E} = -2, $$ since it is topologically a $2$-sphere. Hence $$\int_{E} c_{1}(X) = \int_{E} -K_{X}|_{E} = \int_{E}(-K_{E} + E)_{E} = 2 -1 = 1.$$

Hence the minimal Chern number is $1$. Given any toric surface we can blow up a torus invariant point and the same calculation will give that the minimal chern number is $1$.

So the only remaining cases to check in dimension 2 are minimal models. Since they are rational there is only $\mathbb{P}^{2}$ and the Hirzebruch surfaces $\mathbb{F}_{i}$. The minimal Chern number of $\mathbb{P}^{2}$ is 3.the minimal Chern number of $\mathbb{P}^{1} \times \mathbb{P}^{1} = \mathbb{F}_{0}$ is 2. The minimal Chern number of $\mathbb{F}_{1}$ is $1$. In general, the minimal Chern number of $\mathbb{F}_{n}$ will be $2$ if $n$ is even and $1$ otherwise.