Here we consider the case $k=1$. Then $\text{St}(n,1)=\{x\in\mathbb{R}^n:x^Tx=1\}$ represents the unit sphere in $\mathbb{R}^n$. It follows
$$\text{St}(n,1)\cap\mathbb{R}^n_+=\{x\in\mathbb{R}^n_+:x^Tx=1\}$$
represents the part of the unit circle in the first quadrant of $\mathbb{R}^n$. Let $a^T:=(1,1,...,1)$ define
$$V:=\{x\in\mathbb{R}^n_+: x^Ta\geqslant 1\}$$
Hence the set $V$ represents whatever is in the first quadrant not below the simplex. First we show that
$$\text{conv}(\text{St}(n,1)\cap\mathbb{R}^n_+)=\text{convSt}(n,1)\cap V$$
By definition $\text{convSt}(n,1)=\{x\in\mathbb{R}^n:x^Tx\leqslant 1\}$ is a convex set (the closed unit ball) so is the set $V$. Hence $\text{convSt}(n,1)\cap V$ is a convex set because it is the intersection of two such sets. Moreover $\text{St}(n,1)\cap\mathbb{R}^n_+\subseteq \text{convSt}(n,1)\cap V$ since $x\in\text{St}(n,1)\cap\mathbb{R}^n_+$ would imply $x^Tx=1$ and $1\geqslant x_m\geqslant 0$ for all $m=1,2,...,n$ which in turn yields
$$1=x^Tx=x_1^2+...x^2_n\leqslant x_1+...+x_n=x^Ta$$ and hence $x\in \text{convSt}(n,1)\cap V$. On the other hand $\text{conv}(\text{St}(n,1)\cap\mathbb{R}^n_+)$ is the smallest convex set which includes $\text{St}(n,1)\cap\mathbb{R}^n_+$ thus we get
$$\text{conv}(\text{St}(n,1)\cap\mathbb{R}^n_+)\subseteq \text{convSt}(n,1)\cap V$$
By definition of a convex hull we have
$$\text{conv}(\text{St}(n,1)\cap\mathbb{R}^n_+)\\=\{y\in\mathbb{R}^n_+: y=\sum_k\lambda_kx_k,\hspace{0.1cm}\lambda_k\geqslant 0, \sum_k\lambda_k=1, x_k\in \text{St}(n,1)\cap\mathbb{R}^n_+\}$$
From this definition it follows that the simplex $$V_0:=\{x\in\mathbb{R}^n_+: x^Ta= 1\}\subseteq \text{conv}(\text{St}(n,1)\cap\mathbb{R}^n_+)$$
as well as any line segment joining any two points in $\text{St}(n,1)\cap\mathbb{R}^n_+$. Therefore by construction we get the equality
$$\text{conv}(\text{St}(n,1)\cap\mathbb{R}^n_+)= \text{convSt}(n,1)\cap V$$
So for the case $k=1$ we get the characterization
$$\text{conv}(\text{St}(n,1)\cap\mathbb{R}^n_+)=\{x\in\mathbb{R}^n_+:x^Tx\leqslant 1, x_1+....+x_n\geqslant 1\}$$
For $k>1$ one needs to find an appropriate definition for a simplex. The above idea should work.