Which calendar numbering system did the Franks use?

There are several different aspects in a "calendar". What you ask is not what calendar but what "Era" did the franks use. Eras is the correct concept to refer to year numbering.

What the franks used: Eras (counting years):

• Indictio (rare, tax contexts)
• Metonic Cycle (rare, astronomical contexts)
• Epacta (religious contexts)
• Regnal Years (common)
• Encarnation Years (common late period, by Italian influence)
• Nativity Years (common very late period)

The (Christianized) Julian Calendar by Dionysius Exiguus:

In that we now call AD 532 Dionysius Exiguus proposed as an alternative to other cycles of years for use in fixing Easter dates one of 532 years (see Fotheringham, 1931 and Poole, 1969). He regarded one such cycle as having just been completed, the one beginning in the alleged year of the incarnation of Christ, that is what we call 1 BC. Using the other rules involved in his system, he worked out Easter dates for the first few decades of the second cycle beginning in AD 532 and others later extended the calculations for further periods. In the West the system steadily won favour over others using other cycles, though there seems to have been no intention at first to use it for chronographic purposes. The dating of Easter was one of the many issues on which there were disputes in England which had been christianized from the north by the Celtic missionaries and from the south by the Roman missionaries. At the Synod of Whitby in AD 644, the various issues discussed there were settled in favour of Rome-the dating of Easter, the way to cut a tonsure, the recognition of the primacy of the Bishop of Rome as St Peter's successor and so on. The very learned Venerable Bede (circa 673-735) was familiar with all this and with the Dionysian cycle. Among his works is an essay on the calendar, De Temporis, in which he remarks that some periods are determined by nature (he instanced the year), some by tradition (he instanced the Julian months) and some by authority or arbitrary decision (e.g., the 7-day week).

In his great History of the Church in England, looking for a way of specifying the years he resorted to the Dionysian years after the incarnation of Christ. Indeed, in his second chapter he reports that the preparations for the invasion by Julius Caesar of Britain were made in the six hundred and ninety-third year from the foundation of the city of Rome or the sixtieth year before the incarnation of our Lord.

His convention was taken by the English missionaries into the Frankish Kingdom and later Charlemagne, establishing a new Roman Empire in Western Europe, made it official in the greater part of the West. It spread somewhat later to Eastern Christendom.

Source: Time and the Calendars by William Matthew O’Neil

The only exception is that Charlemagne changed the names for the months:

For until 800 it seems the Franks had called the twelve months by a mix of classical Latin or "barbarous" (and hence pagan) names. Einhard implied that Charlemagne disapproved of such usage, particularly after he took on the imperial office and grew more interested in more h*m*geneous and more explicitly Christian norms and practices across the realm: the relevant section of his biography emphasizes extant discrepancies in customs and Charlemagne's desire for codification and unification. The freshly crowned emperor, Einhard suggested, believed a properly ordered and God-pleasing society should use a single set of vernacular Frankish words for the year's subdivisions, and these labels should reflect eternal truths like the passage of the seasons, Christian belief, and especially peasant labor.

Source: Weeds and the Carolingians by Paolo Squatriti

In The Life of Charlemagne, Einhard, Charlemagne’s biographer said:

He likewise designated the winds by twelve appropriate names; there were hardly more than four distinctive ones in use before. He called January, Wintarmanoth; February, Hornung; March, Lentzinmanoth; April, Ostarmanoth; May, Winnemanoth; June, Brachmanoth; July, Heuvimanoth; August, Aranmanoth; September, Witumanoth; October, Windumemanoth; Novemher, Herbistmanoth; December, Heilagmanoth

Charlemagne’s reforms to the calendars month names were nevertheless futile:

While Charlemagne's calendar reform, the most ambitious in Europe since Caesar and Augustus, had considerable success and can still be discerned behind the current global system for keeping track of time's passage, his attempt to rename the months seems to have persuaded no one. Outside Einhard's biography it left no traces in texts, and people (at least literate people) in the Carolingian empire continued to call the months by their Julian names.

Therefore Einhard’s rationale behind listing Charlemagne’s month names reforms were plausibly to bolster Charlemagne’s achievements and make him look better. But, nonetheless, creating the Charlamagne calendar or Frankish calendar.