...Is there a way of converting the crank without replacing the BB completely?
No: the interfaces between the cranks and BB (ie "square taper" or "cottered") aren't interchangeable.
However, square taper cranks and BB are cheap, plentiful and a useful upgrade over cottered ones.
Just remember to match the front and rear chainlines.
However, square taper cranks and BB are cheap, plentiful and a useful upgrade over cottered ones.
Just remember to match the front and rear chainlines.