The expected data is in fact available on GAP4 (as pointed out in private by Frank Lübeck) with the following command:

For $q$ even:

gap> Print(CharacterTableFromLibrary("SL2even"));

For $q \equiv 1 \mod 4$:

gap> Print(CharacterTableFromLibrary("PSL2even"));

For $q \equiv 3 \mod 4$:

gap> Print(CharacterTableFromLibrary("PSL2odd"));

The quickest way to display the character table for a fixed prime-power $q$ is:

Display(CharacterTable( "PSL", 2, q)

The generic character table is in fact available on GAP4 (as pointed out in private by Frank Lübeck) with the following commands:

For $q$ even:

gap> Print(CharacterTableFromLibrary("SL2even"));

For $q \equiv 1 \mod 4$:

gap> Print(CharacterTableFromLibrary("PSL2even"));

For $q \equiv 3 \mod 4$:

gap> Print(CharacterTableFromLibrary("PSL2odd"));