The short answer is, it would depend. The long answer is: At least DMG (not sure about GBC etc) has protection diodes on the link port inputs which will conduct anything above 5 V plus the diode drop to Vcc. What actually happens depends on the output impedance of the module, and how much current the Gameboy is using from Vcc. Usually, a synth module will have its outputs coupled in series with something like a 1k resistor, mainly to protect from a short circuit if you connect an output to an output.
So that gives you something to do a rough calculation on. 10 V-5 V=5V. 5 V/1000 ohm=5 mA. So, the Gameboy must use (ie be able to absorb) >5 mA to survive. If we go by the 0.7 W rating on the back, it will use maybe a maximum of 0.7 W/5 V=140 mA. Less than that typically, but ballpark figure much higher than 5 mA. Especially with a backlight installed.
Note that this is not guaranteed behavior. A short 10 V pulse edge might still slip through and destroy something. The safety calculation I made above relies on the module having an output series resistor, which it may not have. Using one module might work, but using another might destroy the Gameboy.
What would be better is to make a sync adapter with a series resistor and parallel zener diode to guarantee protection from excessive voltage.
DMG schematic: http://gbdev.gg8.se/wiki/articles/DMG_Schematics