Hmmm, it's certainly grammatically incorrect but I think that looking at the source of the text in question might explain why it's "me" instead of "I".
Urban Dictionary's contributors tend to use a lot of internet/technological slang.
I've noticed recently a trend of "me" instead of "I" especially among younger people who spend a lot of time online, and have traced it back to a couple of chat programmes, most noticeably perhaps Skype, where the person speaking can type "/me" followed by an action and the programme will convert it into the format "Username does x action" - so typing "/me hits you with a trout" would appear in the programme as "Tullia hits you with a trout". I assume this function was designed as a timesaver for those with very long usernames, or usernames with odd symbols in.
It's becoming something I've seen people do on forums now too, even though the function isn't there to convert the text as it is in the chat programmes.
An alternative explanation [and possibly simpler] is that "quoth" was just a typo for "quote". As in "quote me", which makes some form of sense in the context I think, and seems more likely than over-complex attempts to interpret it as an ironic archaism.