You have additional $ signs in your code snippet. The superscript ^, and subscript _ need to be in math mode.
Use $\triangle^m$ for superscript. If you want a subscript use $\triangle_m$.
If you intention is to make the entire term a sub- or superscript then you'll need to enclose it with a {} group.

Note:
- For the superscript version, I think it looks better if you slightly move the superscript to the left via a
\kern.
- If your intent is to use display math you should use
\[ and \] instead of $$ delimiters. See Why is \[ ... \] preferable to $$?.
Code:
\documentclass[fleqn]{article}
\usepackage{amsmath}
\begin{document}
Inline math: $\triangle^m \quad \triangle_m$
%
\[ \text{Display math:}\quad \triangle^m \quad \triangle_m \]
%
If the entire term is to be super or sub script:
\[ \text{Display math:}\quad x_{\triangle^m} \quad x_{\triangle_m} \]
\end{document}
$\triangle$to get the △ shape, but that’s because you need to be in “math mode” to use the\trianglecommand. A single$puts you into math mode, where you can use both the\trianglecommand and the superscript operator. – J. C. Salomon Oct 4 '12 at 12:27