I inherited a home from my mother. The home was sold six months from the time of her death, and no improvements were made on the home. Do I have to pay capital gains tax?

Probably not.

The basis of the house is "stepped up" to FMV as a matter of law. I doubt that the house increased in value enough in six months to generate any capital gains, especially when you add the cost of the sale to the basis.

You WILL have to report the sale on your tax return, but I doubt that any capital gains tax will be due.