I'd like the mountains that I can climb

没有用错.也可以用where,但用where时climb作不及物动词,而用that时climb作及物动词.
用that时climb显然是及物动词,不用多说.
用where时I'd like the mountains where I can climb.=I'd like the mountains which I can climb up.所以climb是不及物动词.此时where=up which.
climb可以做及物,也可以做不及物,你可以查词典的.