You might need to expose the C# code as a .NET library. Check the following links:
https://www.mathworks.com/help/matlab/matlab_external/calling-net-methods.html#bt00pa8-4
In order to control/open a web browser from MATLAB, you can use the 'web' command. Refer to the following documentation:
https://www.mathworks.com/help/matlab/ref/web.html