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.
Firstly, who are the Quartodecimans?
The Quartodecimans are followers of the Early Church who kept Easter on the fourteenth day of the moon of Nisan, whatever day of the week that might be, following therein the tradition which was claimed to have derived from St. John the Apostle.
Irenaeus states that St. Polycarp, who like the other Asiatics, kept Easter on the fourteenth day of the moon, whatever day of the week that might be, following therein the tradition which he claimed to have derived from St. John the Apostle, came to Rome c. 150 about this very question, but could not be persuaded by Pope Anicetus to relinquish his Quartodeciman observance. Nevertheless he was not debarred from communion with the Roman Church, and St. Irenæus, while condemning the Quartodeciman practice, nevertheless reproaches Pope Victor (c. 189-99) with having excommunicated the Asiatics too precipitately and with not having followed the moderation of his predecessors. - New Advent.
The question thus debated was therefore primarily whether Easter was to be kept on a Sunday, or whether Christians should observe the Holy Day of the Jews, the fourteenth of Nisan, which might occur on any day of the week. Those who kept Easter with the Jews were called Quartodecimans or terountes (observants); but even in the time of Pope Victor this usage hardly extended beyond the churches of Asia Minor. After the pope's strong measures the Quartodecimans seem to have gradually dwindled away. Origen in the "Philosophumena" (VIII, xviii) seems to regard them as a mere handful of wrong-headed nonconformists. - New Advent.
As we see the Quartodecimans believed in celebrating the Easter or the Resurrection of Our Lord Jesus Christ on the same day as the Jewish Passover or Pascha which is held on the 14th day of Nisan.
The Council of Nicaea (A.D. 325) fixed the date for the celebration of Easter as the following:
The Sunday to be kept as Easter Day must necessarily occur after the vernal equinox, then identified with 21 March of the Julian year. This was the main difficulty which was decided by the Council of Nicaea.
The Council also Ruled :
•that Easter must be celebrated by all throughout the world on the same Sunday;
•that this Sunday must follow the fourteenth day of the paschal moon
•that that moon was to be accounted the paschal moon whose fourteenth day followed the spring equinox;
•that some provision should be made, probably by the Church of Alexandria as best skilled in astronomical calculations, for determining the proper date of Easter and communicating it to the rest of the world (see St. Leo to the Emperor Marcian in Migne, P.L., LIV, 1055).
As for how often the Eucharist was celebrated in the Early Church is really a separate question in itself. The article entitled "How Often Did the Early Christians Observe the Lord's Supper?" can perhaps help shed some light on this subject.
Acts 2:42 indicates a regular sharing in the Lord' supper, but the question still remains, how often was it observed?
The answer to this question appears later in this same letter (I Corinthians 16:1,2). The Corinthian Christians met on the first day of the week! In fact, this was also a practice among the churches in Galatia, which Paul indicates in I Corinthians 16:1. - How Often Did the Early Christians Observe the Lord's Supper?
Wikipedia explains the Quartodeciman controversy thus:
The Quartodeciman controversy arose because Christians in the churches of Jerusalem and Asia Minor celebrated Passover on the 14th of the first month (Aviv), while the churches in and around Rome changed to the practice of celebrating Easter on the following Sunday. The difference was turned into an ecclesiastical controversy when synods of bishops which held to Apostolic tradition condemned the practice.
Polycarp of Smyrna (c. 69 – c. 155) was one of those bishops that held to Apostolic tradition, claiming to have received his tradition directly from the Apostle John.
Irenaeus, in his letter to Victor of Rome, writes regarding Polycarp's visit to Rome when Anicetus was its bishop (c. 68-153):
Neither could Anicetus persuade Polycarp not to observe what he had always observed with John the disciple of our Lord, and the other apostles with whom he had associated; neither could Polycarp persuade Anicetus to observe it, as he said that he ought to follow the customs of the presbyters that had preceded him.
Although Polycarp and Anicetus were unable to reach an agreement regarding celebrating Passover on Nisan 14 or Easter on the following Sunday, the same letter to Victor states:
Anicetus conceded the administration of the eucharist in the church to Polycarp, manifestly as a mark of respect.
This shows at least three things:
When or how frequently the eucharist was celebrated was not part of the Quartodeciman controversy.
The eucharist and the Passover were distinct celebrations to all parties involved, including Polycarp, a "Quartodeciman" himself.
Polycarp, while adamant about Nisan 14 and the traditions he received from the apostle John, had no qualms with celebrating the Eucharist on another date.