I am new in QT using C++ 98
It's almost 2023. It's time to move on :)
As for the code - it's kinda all over the place.
You're just iterating over some widgets. Why do you even need a signal mapper here? Especially since you map those widget to themselves? That doesn't make sense.
find_child will return nullptr if it couldn't find anything. You're not checking that so you're probably getting nullptr and then trying to call btn->setStyleSheet on it. That will crash.
Those buttons don't seem to be children of the signal mapper. At least there's no code here that would make them that. Your signal mapper doesn't have any children here, so find_child won't find anything and will return nullptr. You should call find_child on the actual parent of those buttons.
In any case, find_child operates on object names, not variable names. Are these widgets actually named with setObjectName()? If not then find_child will not find them and return nullptr.