Hi,
Possibly, but we really need more details, and probably not by using only the "normal" PICaxe Basic commands and User Manuals. Therefore, it's certainly not "easy".
Firstly, the 555 is a moderately "High Voltage, High Current" analogue chip, so can deliver a "continuously variable frequency" (using a variable resistance) up to 12 volts amplitude, or more. The PICaxe (using internal code) can only change the frequency in "steps", with outputs up to about 5 volts and 20 mA.
To get 200 kHz you would need to use a PWM output and the 08M2 has only one (Pin c.2 = Leg 5). For the "Not Q" signal output, you then need to find an on-chip inverter (to invert the signal output on c.2). There are several possibilities, e.g. an analogue comparator or the S-R Latch, but both of these also use (only) the c.2 output pin (or Leg 2, the programming input, which also is not available).
Therefore, the only internal hardware I can think of is the "Data Signal Modulator", which has its output on Leg 7 (the programming output), not very convenient, but "possible". The DSM is not directly supported by PICaxe Basic, so you would need to use POKESFR commands, and reference to the "advanced" Microchip Base PIC data sheet
here.
Cheers, Alan.