According to Den Danske Ordbog, it actually comes from French fol !
"fjollet" comes from the verb "at fjolle" which means to play silly. This verb in turn comes from the noun "et fjols" meaning a fool.
OPRINDELSE via oldfransk fol fra middelalderlatin follis 'nar, opblæst person', det indskudte j i fjols har nedsættende betydning ligesom fx i fjog
Translation: "ORIGIN via Old French fol from Medieval Latin follis 'fool/jerk, self-important person', the inserted j in fjols has a derogatory meaning as in for example fjog.