Can AI (NLP) convert user questions (text) into database SQL queries?
Yes it is possible. You can train end to end an RNN over the training data. It means input the user query as an input and set the output for the sql query.
Definitely the most challenging part is preparing the training set as RNN needs a lot of data to be learned. 

Also, for the training set, it might you can use a GAN‌ to generate more queries for the users' query dataset that you have. Notice that, as you want the query generator for the specific schema of the database, you need your own training data and it could not be generalized over all kinda schemas.