Here's a simple counterexample: Let $d=4$ and $k=2$. Let $X\subset\mathrm{GL}_4(\mathbb{R})$ consist of a single element $J$ where $J^2=-1$. Then $J$ is not conjugate to any upper triangular matrix (over $\mathbb{R}$) since it does not have real eigenvalues. Meanwhile, the eigenvalues of $\Lambda^2(J):\Lambda^2(\mathbb{R}^4)\to\Lambda^2(\mathbb{R}^4)$ are $\pm 1$, and $\Lambda^2(J)$ has a basis of eigenvectors.
On the other hand, if $X$ is a connected Lie subgroup of $\mathrm{GL}_d(\mathbb{R})$ such that $$ \Lambda^k(X) = \{ \Lambda^k(x)\ |\ x\in X\ \} $$ is conjugate to an upper triangular subgroup of $\mathrm{GL}_m(\mathbb{R})$ where $m = {d\choose k}$ and $1< k < d$, then $X$ must, itself, be conjugate to an upper triangular subgroup of $\mathrm{GL}_d(\mathbb{R})$. This essentially follows from the Lie-Engel Theorem, since $X$ has to be solvable, and the reality of its roots in the $\Lambda^k$-representation forces the reality of its roots in the $\Lambda^1$-representation.