Mom has Alzheimer's and told a relative that my Dad and I probably have a "thing"! How do I handle?

That would be embarrassing. If that happened to me I would let the relative know that it wasn't true. She probably knows your mother's mind is not healthy, so will understand. I hope your mother doesn't believe what she told your relative.

Here is a good article: https://www.agingcare.com/articles/How-to-handle-alzheimers-disease-lying-144204.htm