I have only had the form email me, not store anything to the database. I imagine though you need to modify ContactUsSend in the CommonController.cs file.
I have tried figuring it out myself now, but got a bit lost. My idea would be to see how a newsletter subscription is added, as this must work with the database. The controller is NewsletterController.cs, which deals with Nop.Services.Messages.NewsLetterSubscriptionService.cs. This uses class of NesLetterSubscription defined in Nop.Core.Domain.Messages.NewsLetterSubscription.cs. Other than that I'm a bit lost at the moment, sorry.
Maybe someone else can help, or there is another thread on this forum about adding to the database.