Moodle quiz questions can be exported in a variety of ways and then imported into other sites. (See Export questions and Import questions). We have an area on Moodle.net.,our Moodle Community sharing hub where people can share quiz questions. I think there is only one grammar question set there currently, but it would be great if you could share some of yours - and perhaps that will encourage others to share theirs Here is the link: Moodle.net quiz questions database.
Teaching with Moodle
Sharing quizzes of English grammar
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