Is there a certain numbers of years in which the solar year and lunar year cycles (calendar) coincide?
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There really is no such thing as a lunar year. The Moon's period around the Earth is a month, and a lunar month is the time it takes to go from new moon to new moon on the average. This is 29.53058868 days. 235 of these is 6939.6883398 days.
The tropical year, defined as the time it takes for the seasons to repeat, is 365.2422 days, so that 19 tropical years is 6939.6018 years. The difference between this and 235 lunar months is 0.0865398 days.
This is so close that lunar phases will repeat every 19 years. It was full moon when I was born, so when I reached my 57th birthday, the moon was again full. The 19-year cycle is the basis for lunar-solar calendars, such as the Jewish calendar. In a 19-year period, there will be seven 13-month years and eleven 12-month years. The rule for determining Easter also involves a lunar calendar - epacts and the Golden Number (which one of the 19 years we are in), as well as the Dominical letter, determine the Paschal full moon, the first after the equinox, and hence Easter. By the way, the Golden Number of 2007 is 13.
I don't know what you mean by lunar year cycle. The Moon goes around the Sun once a year, just like the Earth. The Sun cycle is not a yearly thing; it has to do with the magnetic field.
The Moon completes one orbit of the Earth in 27.32 days, which does not produce an easily commensurable result with Earth's 365.2422 day orbit of the Sun. There is an 18.6 year cycle which permits eclipses to more or less repeat, called the saros cycle.
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