Click to expand styles? You mean the Quick Styles on the Home tab?
That has nothing to do with Quick Tables.

A Quick Table is a Building Block. You can use styles in a Building Block but it is not a style.

1. You can type the first four unique characters of your Quick Table building block's name and press Enter to insert if you stored it anywhere except the Building Blocks.dotx template.
2. If it is in Building Blocks.dotx, you can, after typing enough of the name to distinguish it, press F3 to insert it.
3. You can go under Insert Table > Quick Tables and pick your table from there.
4. You could assign a keyboard shortcut to it.

• Assigning Keyboard Shortcuts in Microsoft Word 2007-2021 (365)
• AutoComplete - for AutoText and Other Building Blocks - Microsoft Community
• Where can Building Blocks be stored?
• Understanding Styles in Microsoft Word