I can easily copy/paste Chinese letters from for example Google Translate to a Word document, but I am not able to copy to an Indesign file. The specific Chinese font should be available as a standard font on my computer as it only causes problems for me when I am working in InDesign. When I try to write the Chinese letters, boxes with an ? comes up instead.
Okay, you say you copy/paste to Word, but "write the Chinese letters" in InDesign.
I must assume you are pasting to InDesign as well, no?
If so, your Preferences > Clipboard Handling > When Pasting Text and Tables from Other Applications... setting may be set to "Text Only" in which case the pasted text would take on the default font, and not whatever Chinese font you have installed.
Other possibilities might be that there is a Paragraph or Character style, presumably assigning a font other than your Chinese font, already in effect where you're pasting.
You'd have to be certain you have these conditions right before you dig deeper.