Is it harmful to use oil after 1 months and 1 week after hair transplant?

Given the stage your at after hair transplant i.e. 6 weeks there is no issue with using oil. Both organ and coconut oil are fine to use. In fact a lot of my clients at this stage prefer to use oil in order to reduce the itch and dryness that may occur. It helps sooth the scalp. All the best. 

Just guessing, they may have warned you against excessive usage, as no hair transplant clinic with a doctor should tell you to avoid using oil.
But I have seen many cases where former patients used oil in excessive amounts and on top, they weren't washing their hair everyday. So eventually it caused clogging and dried oil on scalp was coming off uncertain times in flakes. This is something you need to avoid and apply the oil only in small amounts.

There will be no harm for using oils as the wound is completely healed already. I also advise my patients to use oil to soften the scabs two weeks after surgery. No worries.