I think they mean to include ("quote") the information about the invoice # and date when making the payment, so they know what to credit the payment towards.
I would expect it to say "please reference" rather than "quote," which is confusing when discussing commercial terms, since a "quote" is an offer to sell.
If you have a question about the English language and would like to ask one of our many English teachers and language experts, please click the button below to let us know: