Does converting adjacency matrix representation of graph of size $n \times n$ to adjacency list always require $O(n^2)$ time?
"Nothing is always."

Without any prior knowledge, the worst case time-complexity of converting an adjacency matrix representation of a graph in 0,1 values to a corresponding adjacency list representation is $\Theta(n^2)$.

However, if we know the graph is a sparse matrix with number of edges $m=o(n^2)$ and its adjacency matrix is stored in some compressed format like Yale format, then we can expect a conversion takes $o(n^2)$ time.