Do voltage converters convert the cycle (Hz)?

No, voltage converters do not convert the cycle - they do however convert the voltage.  Most electronics will function properly with just the voltage conversion.

North American 110 - 120 volt  electricity is generated at 60 Hz. cycles alternating current.  

Most overseas 220 - 240 volts is generated at 50 Hz cycles alternating current.     This difference in the cycles may cause the motor in your 60 Hz appliance to operate slightly slower than when used on a 50 Hz foreign electric cycle.  However most modern electronic equipment including computers, TVs, monitors, game systems, printers,  stereos,  DVD player etc.  will not be affected by the different in cycles.