How can I export hyperlinks to Autodesk?

You can add the hyperlink to the XData of the Eyeshot entities.

Circle c1 = new Circle(Plane.XY, 20);

c1.AutodeskProperties = new AutodeskMiscProperties();
c1.AutodeskProperties.XData = new List<KeyValuePair<short, object>>();
c1.AutodeskProperties.XData.Add(new KeyValuePair<short, object>(1001, "PE_URL"));
c1.AutodeskProperties.XData.Add(new KeyValuePair<short, object>(1000, "https://www.devdept.com"));

design1.Entities.Add(c1);
WriteAutodesk wa = new WriteAutodesk(new WriteAutodeskParams(design1.Document), "out.dwg");
wa.DoWork();