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I've wondered if many of the abbreviations are empty, or if they are similar to the recursive acronym like PHP and GNU. For instance, I learned that CQ mimics the two syllables of a French word for Cequite, which means to secure. So, what does QRZ, and others, stand for, how did they originate?

They were created to save time in Morse code communications.

From https://en.m.wikipedia.org/wiki/Q_code:

The Q code is a standardized collection of three-letter codes all of
which start with the letter "Q". It is an operating signal initially
developed for commercial radiotelegraph communication and later
adopted by other radio services, especially amateur radio.

Although Q codes were created when radio used Morse code exclusively,
they continued to be employed after the introduction of voice
transmissions. To avoid confusion, transmitter call signs are
restricted; no country is ever issued an ITU prefix starting with "Q".

Here is a list of them.

Here's a quote from https://en.wikipedia.org/wiki/Q_code .

The original Q codes were created, circa 1909, by the British government as a "list of abbreviations... prepared for the use of British ships and coast stations licensed by the Postmaster General".[citation needed] The Q codes facilitated communication between maritime radio operators speaking different languages, so they were soon adopted internationally. A total of forty-five Q codes appeared in the "List of Abbreviations to be used in Radio Communications", which was included in the Service Regulations affixed to the Third International Radiotelegraph Convention in London (The Convention was signed on July 5, 1912, and became effective July 1, 1913.)

It is my understanding that the 'Q' was used as there were very few words that began with that letter , thus making it easy to discern by ear. Many of the 'Q' codes are pairs of interrogative and statement functions. Example , issuing a QRO is either asking the recipient to increase power or stating that the sender will increase power. Another , issuing QTH ( which follows from "Question The House" ) is either what is your home location or this is my home location. In all four cases , what follows the code determines the intent.