Please go to Insert > Special characters. Type the word "cent" in the search field. You should see the symbol in the left panel. Click on that and it will be inserted into your document.
Ava, this thread is about Google Docs, not Sheets. Sheets and Docs don't have the same functions or capabilities, and this feature isn't available in Sheets.
You can use the CHAR function to produce the symbol like the cent...
=CHAR(162)
...to show it after a number...
=99&CHAR(162)
99¢
You can also use the numeric keypad if you have one and then while the cell you want the symbol in is in editing mode (cursor is blinking in cell) then hold down the ALT key and type 155 on the keypad.