According to Deadline the season 3 renewal came in at the last minute, because it was considered an international success for ABC.
It came down to the wire, but Quantico has scored a last-minute
13-episode third-season renewal from ABC less than 24 hours before the
network’s upfront presentation. When it comes back, the terrorism
drama starring Priyanka Chopra will have a new showrunner at the helm
as creator/executive producer Josh Safran is stepping down from the
role. Safran will stay on as a consultant for Season 3. Search is
underway for his replacement.
Quantico, from ABC Studios and producer Mark Gordon, was a breakout
when it debuted in fall 2015, breaking delayed viewing lift records.
But with its dense narrative and heavy serialization, the series
started to lose momentum in the second half of its first season and
couldn’t stop the ratings declines with a season 2 reboot. In its
second season, Quantico has averaged a 1.3 adults 18-49 (Live+7).
Still, the series has a global star, Chopra, as its lead, and has been a strong international seller for ABC Studios, which helped for the renewal.
For an unspecified reason, its previous showrunner and executive producer Josh Sefran stepped down and was later replaced.
As mentioned in the Deadline article, ABC seemed to believe that the reason season 2 dropped in the ratings, was because the show was too serialized (It's no secret that the networks often believe that an episodic structure works better for allowing audiences to be able to "jump into" the series at any time and often avoids deeply serialized structuring.)
So it seems like that has to be reason why it not only recieved a shorter season (to meet budget concerns, as they were also debating moving production), but also because they often believe serialized television is not what their demographic wants and thought this was proof of such a theory.