Request can be used as a noun or a verb. Both the noun and the verb are used in relations to asking someone for a favor or asking for something to be given or done.
EX: (as a noun) To obtain information, you need to send in a request.
(as a verb) I requested Mrs. Smith because I hear that she is a good teacher.
A query is a question or inquiry. It's more open ended and used to express doubt or uncertainty.
EX: (as a noun) Do you have any queries about my explanation?
(as a verb) I queried her statement.
So, in general, a request implies a favor and a query is to question or express doubt. Hope that helps!