Second post: is she tattooed? Because given how many "Quarter Horses" are 80% TB anyway, I wouldn't go on "looks like a TB to me" alone. You can always contact the Jockey Club and ask, but I don't think they could do a whole lot, especially not sorting out individual bloodlines (DEFINITELY not enough to get you any papers unless they happened to hit on a proof-positive match for that horse.) The whole testing process would be pricey and then on the extremely off chance it did result in IDing the mare, getting any lost papers would likely end up being expensive/impossible anyway.
IIRC they started phasing it in around 1997/98 and it became a requirement around 2000 (maybe '01?). Not all mares will be DNA typed. Again, if memory serves me correctly, all mares that weren't phased in had to have hair sent in along with the 2000 (or '01?) foals for those foals to be registered. Any mares that were removed from breeding programs prior to that point probably weren't dna typed.
eta - fyi the cost for dna typing through TJC is $80.