You should not. Nonce is used to protect against cross site request forgery attacks (CSRF) in which another aite tries to trick you into submitting a form to your site which will perform some hostile action.
Nonces are unique value that can be generated only by a specific site at a specific time and therefor can not be guess by the attacking site. What you do is generate a nonce and add it to your form when you generate the HTML for it and validate that it is actually was generated by your site at a reasonable time frame before attempting to handle it.
As contact form should not do anything more then store info in the DB or send a mail there is no point in protecting them agains CSRF. for 99% of the time it will make no dfference for the user but as konstantin points out if caching is used somewhere either by a caching plugin or some caching server out of your control (most ISPs probably has one) you might supply to the user a stale nonce which will fail validation and prevent users from submitting the form.