have no objection to + sth / doing 意思是不反对.这是固定搭配,to在这是介词,并不是to do 搭配.
这句话意思翻译过来就是：我不防再听听你的故事.例句
As for me, I have no objection to it.
至于我，我不反对

I have no objection to spending the weekend with them .
我不反对同他们共度周