There are QBasic clones out there for Windows (Liberty Basic is a popular one if I recall correctly). So it isn't unheard of to port QBasic.
Didn't the C64 and Atari ST have BASIC interpreters? Why not just use one of them?
Every computer system has had a version of BASIC. Even the Apple had it's own version (although it was really a part of the Pro-DOS operating system, and there was no IDE, it was all command line). There was even a BASIC cartridge for the Atari 2600.
The only problem with all of these older versions is the fact that they have no built in graphics functions. Graphics were done with PEEK and POKE commands, which directly control the video memory. They are not easy to use by any means, and it was a rare for a noncommercial game to use anything more than ASCII art.
Also, the various versions of BASIC are not very compatible at all. The simple commands like PRINT and GOTO work well enough, but after that, things get questionable. Things like graphics and arrays aren't handled the same, for instance.
Even the changes from ABasic to QBasic were enough that there was a section in the QBasic help files about converting your old programs.
Not to say you can't convert an older BASIC program to QBasic, but there is really no reason to. QBasic is the best and most popular BASIC compiler ever made, so there is little reason to even support other variations.
Also, there would be no way to write a program using the GP32's buttons, so the programs would need to be written and compiled on the PC.
Come to think of it, maybe you don't even need to port QBasic over to the GP32. Perhaps QBasic's compiler (the actual BASIC compiler is a separate program in QBasic) could be replaced with one that creates FXE's instead of EXE's (which of course are designed for GP32 hardware)?