InDesign 2024 will not save with newly installed Monterey 12.7

Try to create a new user account and work there. Will you have the same issue?

We need the rest of your system specs. RAM, especially.

But, if you're working in a Dropbox, OneDrive, or Google Drive folder make sure that any smart sync feature is turned off.

Does your system meet the min requirements?

https://helpx.adobe.com/ie/indesign/system-requirements.html

There are some things you can try - since it's all files and not just one file.

Try resetting your preferences:

Windows: Start InDesign, and then press Shift+Ctrl+Alt. Click Yes when asked if you want to delete preference files.
macOS: While pressing Shift+Option+Command+Control, start InDesign. Click Yes when asked if you want to delete preference files.
A dialog will appear asking to delete the InDesign Preferences, select 'Yes' on this dialog.
Further info and instructions here if needed:
https://helpx.adobe.com/in/indesign/kb/indesign-preferences-support-file-locations.html

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Reinstalling rarely fixes everything - you can do a complete reinstall using the
Creative Cloud Cleaner Tool https://helpx.adobe.com/creative-cloud/kb/cc-cleaner-tool-installation-problems.html

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All else fails - install an earlier version of Indesign

Rollback to a previous InDeisgn version

https://helpx.adobe.com/download-install/using/install-previous-version.html#:~:text=Open%20the%20Creative%20Cloud%20desktop,down%20list%2C%20select%20Other%20Versions.