I think a common interface between the two objects would be a good idea.
Perhaps define the uncommon fields as nullable and have the screen check for null and determine how to display those fields.
int? OrderNumber {get;set;}
DateTime? QuoteDate {get;set;}
EDIT: in response to JC's comment
Perhaps consider trying to take it a step further and, at least logically, consider an Order and a Quote to be the same kind of object but at different stages in the "Order lifecycle". I.e. a "Quote" is an order at the beginning of the lifecycle, whereas an "Order" is an order at the middle or end of the lifecycle.
You could have an OrderState property defined on your interface, and then your UI could use the OrderState property to decide how the quote/order should be displayed, rather than checking each piece of data individually.
If you feel that the problem is more that you have too many if statements in your UI, then perhaps consider creating small user controls to handle displaying chunks of the UI for either a quote or for an order. You could then either dynamically add the appropriate control (a quote control or an order controL) to your UI, or have both controls already added and just show/hide them as appropriate. I would caution, though, that this sounds like it could be a messy approach, so just be careful that the solution doesn't end up being more complicated than the problem that you're trying to solve.