I don't believe stack overflow can be effectively trapped (note the GoSub system is independent). On my system I just get a segmentation fault, Windows will probably have a similarly-fatal error.
It's also important to note that the stack size can differ between computers, so what works for you might not work elsewhere.
Still, let's assume a 1MB stack (not unreasonable). There's a bit of space taken at the start for bookkeeping and runtime stuff. A subroutine call takes 8 bytes for the return address, 8 bytes for bookkeeping and then space for local variables. If we have none, that's 16 bytes per stack frame which gives up a maximum call depth of about 65 thousand (actually 2^16). I haven't tested this, and I might have missed some extra local variables that are implicitly created by QB64, but hopefully it gives you an idea of the kind of scales to expect.