First of all they can both be nouns and verbs. E.g: I requested a refund. We received a request for a refund. I queried the price. We received a query about the price. To request something is to ask for something, an object or a favour etc. A request is a polite demand. To query is to ask about something i.e. you are wanting information. A query is a question.

We want our money back i.e. we are asking for an action to be performed: I requested a refund. = I want my money back We received a request for a refund. = Someone wants their money back. If we change request to query the meaning also changes: I queried a refund. = I asked if the amount refunded was correct. i.e. I think I should have been given more. We received a query for a refund. = This is just incorrect English. We received a query about a refund. = Someone is asking for information about a refund that has happened. i.e. How much was it? We are wanting to know about the price i.e. we are asking for information: I queried the price. = I am not sure the price is correct, I want it confirmed. We received a query about the price. = Someone asked for some information about the price. e.g. Is it cheaper than last month? If we change to request the meaning also changes: I requested the price. = I do not know the price, I want someone to give me the price. We received a request about the price. = This is just incorrect English. We received a request regarding the price. = Someone has asked for a discount. Hope that helps make it clearer!

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!