Assuming this really is a 110V AC unit, which was originally hooked to a 20 amp circuit breaker, and assuming that that your service will support the additional load in your breaker panel, then yes you can replace a 15 amp circuit breaker with a 20 amp circuit breaker. (Be sure to kill the main power coming into the breaker panel before working on the breakers themselves. You probably have 200 amps coming into the panel and an errant screwdriver will kill you.)
Of course, you are also assuming that the wiring for the AC unit is the correct gauge. For 20 amp, the wire for the circuit should be 12 gauge, not 14. If this is newish construction, and in the US, the wire will typically be yellow, not white and you can read the gauge on it. If you don't know for sure, measure the diameter of the conductor (copper wire without the insulation). 12 gauge wire is 0.08 inches in diameter. 14 gauge is 0.064.
If it is 14 gauge wire, and again assuming the US, it is against code to install a 20 amp breaker, and more importantly you run the risk of a fire to do so. In that case, hire an electrician to pull the correct gauge wire.
Lastly, I would inspect the AC unit itself and make sure it is really just 110V/20 amp.