I don't think it would be a good idea to change the color of the bar just to encode whether it's in an even or odd row.
There are probably better ways to accomplish what your client wants.
Here is a good resource prepared by Jonathan Drummey that talks about conditional formatting in Tableau:
If you post your visualization, other people can recommend good ways to accomplish what you are trying to do.
That said, I think this pretty much does what you are asking for: