Neither sentence is right. You need a preposition, either 'in' or 'with', before 'my rent'. The most natural-sounding phrase for me is 'I am six months behind with my rent.'

I have fallen behind (in) my rent for six months I am six months behind (in) my rent (payments). You aren't actually behind your rent - you are behind in your rent payments. You can leave out the 'payments', as that is understood but you need the 'in'.

When written both look fine to me but if spoken I would go with the second one