You can adjust your document settings to stop Grammarly from checking text that appears inside quotation marks. At this time, this setting is available only in the Grammarly Editor.
To prevent Grammarly from offering suggestions on quoted text, click the Menu icon (shown as three horizontal lines) in the upper-left corner and select Editor settings. In the window that opens next, switch on Don’t check text inside quotes:
When you activate this setting, Grammarly will not make spelling or grammar suggestions for quoted text. However, this text will still be checked for plagiarism if you activate the plagiarism checker.