Would it be considered polytheism (shirk) to memorize the Qur’an in exchange for not paying money, given that the motivation is not paying, rather than seeking the pleasure of Allah?

The basic principle is that this act falls under the category of obligating oneself to obedience, which is permissible.

If what you committed to with your friend was merely a promise to obligate yourself, then there is nothing upon you. However, if you intended it as a vow (nadhr) or if it was expressed with the word "covenant" (ahd), then it is an "obstinate vow" (nadhr lajaj), and you have the choice between fulfilling it or offering an expiation for an oath (kaffarat yamin).