Under the Application Styles menu in the System Settings application there is the Widget Styles sub-menu. In this sub-menu I have three options: Breeze, Fusion, and MS Windows 9x. Is there anyway to install more to choose from? If so please give me directions to a website from which to do so.

Those styles (style engines in some cases) are plugins written in c++. The available options are pretty slim, and most distros only bundle that basic set that you see. There are a couple of alternatives available for qt5 with varying levels of configurability:
* QtCurve: (quite likely to be available in your distro's repositories)
** Long history of support and usage with lots of themes available in store.kde.org, provides a fully-compatible gtk2 theme engine
* Kvantum: https://store.kde.org/content/show.php/ ... ent=166241
** fully svg themable style engine with a growing variety of themes available in store.kde.org
* QSvgStyle: https://store.kde.org/p/1162948/
** similar to Kvantum, fewer themes, less active
* Virtuality: https://store.kde.org/p/1005390/
** Exotic and fully flat theme (even more than breeze)
* Style Project (dsp): https://sourceforge.net/projects/styleproject/
** Highly experimental style engine which provides a maclike experience plus a myriad configuration options

And don't forget "Oxygen" that was default in KDE4.
This is still released together with Plasma5, the package is probably called oxygen5-style or similar...

Thanks airdrik for the exhaustive list.
Team_Linux, please notice that Kvantum and QSvgStyle need a lot more resources than native (c++ plugins) styles

Thank you everyone for your feedback as it was very helpful. It makes me glad to see such a vibrant community working together to help one another. I have found my answer plus many other pieces of helpful advice due to the excellent replies given to my question. Thank you all very much for your time and I appreciate the friendliness with which people responded.