Did the Quartodecimans perform the eucharist only once a year on Nisan 14? The Quartodecimans held a Christian 
Passover on Nisan 14. Does this mean that they performed the eucharist (Lord's Supper) only once a year on Nisan 14, or was the Christian Passover something additional to a regular eucharist celebration?
No, "they" celebrated the Eucharist throughout the year.

I am not sure what your source material is, but perhaps there is some confusion about the term "Passover" in a Christian context.  "Easter" has always been referred to using the Hebrew word for Passover - "Pascha" - in the Christian east and continues to be referred to as such to this day in the Orthodox Church.

I put "they" in quotes above, because I don't think "Quartodecimians" refers to a distinct Christian sect, but rather those ancient Sees who had chosen to observe Pascha on Nisan 14 (see, e.g., Eusebius), which was forbidden by the Apostolic Canons (Canon VII)

James Campbell has written a very detailed essay on the history of the dating of Pascha (Easter) that discusses the above and more.