There are a good few form building plugins out there but the best by far is Gravity Forms
Well worth the license fee and you can offset that cost when you bill your client. They will be able to create as many forms as they need then and attach them as widgets in a sidebar or in a blog post or on a standard wordpress page.
In your case you need to see what the form should look like or at least what fields it needs and then you can create the ‘Request a Quote’ page and attach your form to it.