Do I need mortar between 1x6 deck boards and cement board?
A membrane won’t be adequate. If you want ‘waterproof,’ think shower pan. Not literally but by making this area semi-vitreous, there must be a drain or drains and a slope. 

If you’re trying to avoid the disaster of the previous owner, I wouldn’t recommend the travertine, unless you put in a drain. 

Have you considered Saltillo tile, limestone, or another type of stone? You can make wonderful, colorful installations indoors and out. 
 

Regardless of materials used, the water will have to go somewhere and if you post some photos of the area, better suggestions can be made to make a beautiful and practical new flooring solution.