Can I replace a 15A breaker with a 20A breaker?

If your entire branch circuit is using 12 AWG wire, then yes. Swap the old breaker with a new 20 Amp GFCI breaker.

You may upgrade the breaker to 20 A if the wire is 12 AWG copper, but you may not if the wire is 12 AWG aluminum per Table 310.15(B)(16) (quote is from NEC:2011) -- Note, aluminum wiring for 12 AWG is uncommon in homes constructed within the past 30 years:

You will also need to upgrade the outlet if it is the only receptacle on the circuit per 210.21(B)(1), and if the pool motor does not meet the definition of 438.81(B) (quotes are from NEC:2011):

210.21(B)(1) Single Receptacle on an Individual Branch Circuit.

A single receptacle installed on an individual branch circuit shall have an ampere rating not less than that of the branch circuit.

Exception No. 1: A receptacle installed in accordance with 430.81(B).

Exception No. 2: A receptacle installed exclusively for the use of a cord-and-plug-connected arc welder shall be permitted to have an ampere rating not less than the minimum branch-circuit conductor ampacity determined by 630.11(A) for arc welders.

438.81(B) Portable Motor of 1⁄3 Horsepower or Less.

For a portable motor rated at 1⁄3 hp or less, the controller shall be permitted to be an attachment plug and receptacle or cord connector.