You can adjust your document settings to stop Grammarly from checking text that appears inside quotation marks.
To prevent Grammarly from offering suggestions on quoted text, open any document and click the gear symbol in the lower left corner of the page. In the window that opens next, toggle the switch next to 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.