The rental company should have a good claim  to the amount due under the quote. A contract does not have to be a single document titled "contract" and signed by hand by both parties; whether, a contract is formed whenever there is demonstrated agreement by the two parties to the same terms and an exchange of consideration (or something of value), and agreement can be shown by actions as well as by words. Here you have words that show an agreement to the terms, even if there is no signature, per se; and you have the act of accepting the equipment and paying for it for two weeks, which is 100% consistent with there being agreement to the quote's terms (and which is inconsistent with there being no agreement). Under these circumstances, you should be able to show there was in fact a contract or agreement, and therefore you should be able to enforce it in court.