The Classic period Maya had a number of calendrical cycles, but they probably didn't obsess about them as cycles any more than we think of our own repeating cycles of days and months and years. Recent scholars have openly acknowledged that past Maya archaeologists were often more obsessed with time reckoning than the ancient Maya themselves were.
One of their primary cycles was the 260 day cycle that combined 13 numbers with 20 named days (like we cycle the numbers 1-30 with seven named days) that roughly approximates 9 months. Modern scholars have nicknamed it tzolk'in since we don't know the ancient name for this cycle. According to Mark Wright, this cycle may be attested to in Omni 1:21 when it states that Coriantumr dwelt with the people of Zarahemla for "nine moons", which is a fitting translation since there is no English equivalent. What they were saying was "Coriantumr dwelt with the people of Zarahemla for one cycle of the Sacred Round". It is the only time a moon count is used in the Book of Mormon.
The most important Maya cycle was the katun, or period of twenty years. They would celebrate half-katuns of 10 years and quarter katuns of five years, called "hotun". Rulers would typically erect monuments every five years to celebrate the "Period Ending", or even "prophesy" five years in advance that on such-and-such a date the hotun would end and a new cycle would begin. For Mark Wright, the hotun cycle is attested in Helaman 14:2 when Samuel the Lamanite prophecies that Christ would be born in five years with wonders in heaven given as a sign, which ushered in a new period of Nephite history; indeed, they even began to reckon time from the time the sign was given.
Many others have pointed out that the baktun, or 400 year cycle, is attested to several times in the Book of Mormon (Alma 45:10; Helaman 13:5,9; Mormon 8:6). Mark Wright thinks the baktun and the katun are both attested to in Moroni 10:1 when Moroni states that "more than four hundred and twenty years have passed away since the sign was given of the coming of Christ". What he finds most interesting in this verse is that Moroni specifically says he wants his brethren the Lamanites to know that it had been four hundred and twenty years, because that number would carry much more meaning to the Lamanites than it would to the Gentiles.
The Maya "Long Count" records the elapsed number of periods of 400 years + periods of 20 years + years + periods of 20 days + days since the "creation" day of 13 August 3114 BC (although it's unclear exactly what happened on that day; the accounts differ from site to site). Moroni is basically giving an abbreviated Long Count date of 1.1 (1 period of 400 years + 1 period of 20 years).
It should also be noted that Charles Anton, in describing the transcription of the characters which he received form Martin Harris, said "the whole ended in a rude delineation of a circle divided into various compartments, decked with various strange marks, and evidently copied after the Mexican Calender given by Humboldt, but copied in such a way as not to betray the source whence it was derived." Charles Anton may not have believed in the Book of Mormon but he, unwittingly, was the first to connect it with Mesoamerica.
All but the last paragraph of the the above was redacted from something Mark Wright once wrote on the mormondialogue message board.