Which 122x32 display? What interface? Graphic LCD displays come in a few "flavors" in terms of basic interfacing.
Some don't have graphic controllers; they only have segment drivers. These have registers that need to be continuously updated or refreshed. Their timing synchronization requirements would make them difficult to interface with micros like Picaxes.
Some displays have graphic controllers that manage the LCD refreshing tasks. These typically have 8-bit data interfaces (+ other control lines). Google datasheets for chips like the SED1520, SED1335 or T6963C. Capabilities and features vary but pixel-level control is normally the minimum. Some of the Picaxe chips could run these displays.
Finally, there are graphic displays with serial interfaces. Or rather, other people have developed serial 'backpacks' that translate serial commands to the parallel interface of the graphic controllers. The may also add features like multiple graphics pages, drawing commands and custom character sets. These are very easy to control. Even the 08M could draw stuff on a serially-interfaced display.