A star edge coloring of a graph \(G\) is a proper edge coloring in which every path or cycle on four vertices contains at least three distinct colors. The star edge chromatic number of \(G\), denoted by \(\chi^\prime_{\mathrm{st}}(G)\), is the minimum number of colors required for such a coloring. In this paper, we study the star edge chromatic number of several graph classes derived from the Knödel graph \(W_{\Delta,n}\). In particular, we determine bounds and exact values for \(\chi^\prime_{\mathrm{st}}(G)\) for the Knödel graph \(W_{\Delta,n}\), its middle graph \(M\!\left(W_{\Delta,n}\right)\), and its Mycielskian \(\mu\!\left(W_{\Delta,n}\right)\). Furthermore, we provide explicit constructions of star edge colorings that attain these values.