Capitalize when it's a proper noun:
* Library Report (referring to a specific report with a title like "The Library Report 2023").
* The Library Report (when referring to a particular report, often used as the title).
Don't capitalize when it's a general term:
* library report (referring to any general report about a library).
* a library report (referring to one report about a library).
In conclusion:
* Capitalize "Library Report" when it's a specific report with a defined title.
* Don't capitalize "library report" when it's a general term.
If you provide more context about how you're using the phrase, I can give you a more precise answer.