I don't think so. I have never used Groupon, but am I right in thinking you buy a coupon then deal direct with the company named on the coupon? So you would be booking direct with Marriott with a Marriott Groupon anyway, so the best rate wouldn't apply.
Their terms and conditions state "Most Internet Web sites are included in Marriott's Look No Further Best Rate Guarantee (for example, Hotels.com, Expedia.com, Travelocity.com). However, the Web site must be one in which you can complete the transaction on the site. "
As you cannot complete the booking on the Groupon site, I don't think it will qualify. But I would be happy to be proved wrong.
I have used the Best Rate Guarantee once in the past, when I found a very much cheaper rate on Bookings.com (I think their website was wrong, and was still showing winter offer prices for my chosen hotel, which was to my advantage). I booked on Marriott, immediately put in a claim form, and the Booking.com rate was honoured and reduced by 25%. So I think Marriott are fair about their guarantee.