Don’t think it’s possible, it definitely wasn’t when I tried this package some time ago.
A more flexible alternative is https://github.com/invenia/NamedDims.jl + https://github.com/mcabbott/AxisKeys.jl. The former provides named dimensions, which is exactly what you want. The latter gives custom names for rows/columns/…, and these packages are designed to work either by themselves or together.
Your example would look like NamedDimsArray([1 3; 2 4], (:Rows, :Cols)).