Do Google Photos images take quota when they appear in the Google Photos folder in Google Drive?

No, they will not occupy additional storage. You can verify this by selecting the photo in Google Drive, clicking on the "i" at the top right and showing the details of the file. There, the "used storage" will be displayed, which should be "0 bytes" if it was uploaded with the compressed quality settings in Google Photos. Here is an example screenshot:

Free storage from Google Photos