Is it possible to for the mechanical workings of a watch to be changed such that it ends up showing Mars time?
Is the answer same for both a quartz and a mechanical watch?
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Based on CuriousOne's comments, and on Timekeeping on Mars
The average length of a Martian sidereal day is 24h 37m 22.663s (88,642.66300 seconds based on SI units), and the length of its solar day (often called a sol) is 24h 39m 35.24409s (88,775.24409 seconds). The corresponding values for Earth are 23h 56m 4.0916s and 24h 00m 00.002s, respectively. This yields a conversion factor of 1.0274912510 days/sol. Thus Mars' solar day is only about 2.7% longer than Earth's.
A convention used by spacecraft lander projects to date has been to keep track of local solar time using a 24-hour "Mars clock" on which the hours, minutes and seconds are 2.7% longer than their standard (Earth) durations. For the Mars Pathfinder, Mars Exploration Rover, Phoenix, and Mars Science Laboratory missions, the operations team has worked on "Mars time", with a work schedule synchronized to the local time at the landing site on Mars, rather than the Earth day. This results in the crew's schedule sliding approximately 40 minutes later in Earth time each day. Wristwatches calibrated in Martian time, rather than Earth time, were used by many of the MER team members.
With of the quartz clock, you could adjust the stepping motor to coincide with the length of a Martian day.
From Quartz Clock
Battery provides current to microchip circuit
Microchip circuit makes quartz crystal (precisely cut and shaped like a tuning fork) oscillate (vibrate) 32768 times per second.
Microchip circuit detects the crystal's oscillations and turns them into regular electric pulses, one per second.
Electric pulses drive miniature electric stepping motor. This converts electrical energy into mechanical power.
Electric stepping motor turns gears.
Gears sweep hands around the clockface to keep time.
For mechanical clocks, again, as suggested above, an alteration in the gearing could allow for Martian day lengths.