Semen infected with chlamydia comes in contact with non-bleeding cut. will infection be transmitted? or does it have to be mucus membrane contact?

No risk, but...: Chlamydia is not known to infect the skin or cuts. No risk and no worries from such an exposure. Of course anybody with known or suspected chlamydia should not be having sex or taking any chance of exposing anyone else, including contact of his semen with skin or anywhere else. That includes non-intercourse contact, e.g. hand-genital or oral sex. He and his sex partners need treatment ASAP.