I just put window air conditioners in my boys bedrooms. The 20 amp breaker that the two rooms are using keeps tripping. Will installing a 30 amp breaker solve the problem?

Nope, that would create a fire hazard, the 20 amp breaker is tripping because the circuit is being overloaded. Installing a higher amperage breaker would defeat the safety.

You might try having an electrician run a dedicated circuit for each unit if they are pulling that much of a load.

You must know the gauge of the wires on this circuit to give a definitive answer. In all likelihood it would be a fire hazard since electricians would not use a wire heavier than 12 gauge for a 20 amp breaker. This chart should help you understand the correct relationship between wire gauge and breaker size. Wire sizes, gauges, for breakers (http://www.greatinspector.com/faq-elec-wiresize.html)

An Air Con system will generally pull 3 times the current on start up, This can be overcome by fitting a type D breaker which is designed to absorb this start up current. Never just put a bigger ampage of breaker in as this will only make any fault go away until the fault is big enough to trip the bigger one but if the wiring can't take it then it will be too late.

Wiring genius, type D breaker is not used in North America in standard residential wiring.