Nope, those are two completely different things. Basically, being in a weapon group doesn't change anything about the weapon, it only alters how the weapon interracts with feats or class features that explicitly use weapon groups.
Note: if the word "group" doesn't appear in the description, than whatever it is doesn't use weapon groups. In this case, Flurry of Blows doesn't mention "group", and thus doesn't care about them. Indeed, there're three weapons in the monk group that lack the monk weapon property (Tri-point Double-edged Sword, Urumi, and unarmed strike), and three weapons with the monk weapon property that aren't in the monk weapon group (Knuckle Axe, Temple Sword, and Waveblade).
Versatile Design helps to flurry the weapon, but you'd need Ascetic Style and Ascetic Form. Plus Exotic Wepaon Proficiency, of course.
Was having a rules disagreement from one of the players in my living world that it automatically grants the quality because the weapon group mostly grants it. Trying to be nice, but don't rules lawyer a rules lawyer on his own server.
This is pretty much the opposite of a rules lawyer. This is a case of someone making something up, and claiming it was a rule.
The whole thing is based on a faulty assumption. "the weapon group mostly grants it" is an objectively false statement - the weapon group doesn't grant anything. That most weapons in it have the monk property doesn't come form the group, but from each weapon's individual weapon list entry. Anyone claiming otherwise shall quote a rule.
Just in case, here's some more evidence:
• The term "monk weapons" is listed in the index of the CRB (pg. 574), with two entries: The first is in the Monk description, the other in the rules for weapon special features. Neither contains the word "group":
"" CRB pg. 57
"" CRB pg. 145
• If "monk weapon" meant "weapon in the monk weapon group", you could flurry with any weapon in the monk weapon group, and the monk special weapon quality would be meaningless.
• The Monk description doesn't reference weapon groups, even though it's the only thing in the CRB interacting with "monk weapons".
• If the weapon group was meant by that term, the Flurry description wouldn't need to say "unarmed strike or a special monk weapon" as unarmed strikes are already in the monk weapon group.
• When they updated that line for unchained, they changed it to explicitly say "unarmed strikes and weapons that have the monk special weapon quality".
• There's exactly one page in the CRB where the term "weapon group" appears, and that's the Fighter description. The idea that a monk player must read the rules for another class to understand how he can use the Monk's main class feature, without even as much as a single reference, is ridiculous.
For reference, all weapons on the monk weapon group list are monk weapons except the tri-point double-edged sword and the urumi. The only weapons that aren't monk weapon group weapons but are monk weapons are the knucke axe and the waveblade, of which only the waveblade was released after Ultimate Combat (in 2017 instead of 2011).
Yeah, kinda sucks that you cannot mod a weapon to add the Monk property to any weapon...