I want to add a column of cells together only if the particular cell is colored green. Can I do this using conditional formatting on a google spreadsheet?
No, Conditional formatting in Google Spreadsheets can only edit text and cell background color. You cannot use it to edit cell values. As Brian P pointed out you can use scripts to do it.
Yes, you can do this with a script (Tools >> Script editor) by using getBackgroundColor() in the Spreadsheet API.