Silpion

This plugin gives a possibility to manage Frequently Asked Questions as a custom post type. Site admin is able to add and edit questions and answers, assign them to categories and then display the output on any page using [faq] or [faq category={{category_id}}] on any WordPress page.
You can embeed the FAQ directly in your code using lcb_faq_display() function.

LCB FAQ

Displays questions with answers with FAQ category column.

Timo Schiller

Kilomoana

This Plugin allows you to include a simple Textbox through which your site visitors can ask you short questions. 
You can answer them through the backend.
See it in action at: (http://kilo-moana.com/ask-me-wordpress-plugin/)

Ask Me

Great !

Simple Plugin to let your users ask questions.

LCB FAQ vs Ask Me