In Calc you would Insert a shape to gain access to the Area dialog’s color tools, as noted there are a few places in the UI that do not expose the Area dialog directly.
Otherwise the default set of color palettes can not be edited within the UI including the standard pallet that most documents will open with by default. These LibreOffice provided program pallets are located in the $TARGETDIR/share/palette directory. You can edit them externally if needed.
So, on any of the color pickers (e.g. Standard toolbar → Font color) use the drop list and select either “custom” or “Document colors”.
You can assemble a palette as “Custom” using the Area dialog’s Color panel, selecting the “Custom” palette from the drop list and using the RGB values, Hex code, or “Pick” color widget to set a color and the"Add" button to add it to the custom palette. The “custom” palette will save into your user profile. Note that all the sRGB colors in the user profile are recorded as large decimal numbers rather than the more compact HEX sRGB value.
“Document colors” are named with document relative colors by HEX value within the document as ODF stanzas on save.
While functional, the “Custom” palette recorded into user profile is tedious to create, and vulnerable to profile reset. Suggest folks with large custom palette copy out the --/org.openoffice.Office.Common/UserColors-- stanzas from their user profile, that is open their profile registrymodifications.xcu in a text editor. Those backed-up stanzas can be appended to a renewed profile when needed to restore their custom palette.
Alternatively you can assemble more ambitious palettes with a text editor–the XML is a very simple color name and color value in HEX–and then written into the share/palette directory as a .soc/.sog file. Then in LibreOffice the newly added palette will show on all the color droplists with the name used as labeling. Be aware that the color names are not written into the ODF documents just the HEX value.
For the motivated, you can distribute palettes to others with a document, and even incorporate into a distributable extension if more widely needed.