Yes, you can, but it's quite hard to do according the my local GUM. Gonorrhea is much more likely. Either way you shouldn't be doing OWO if you're having unprotected sex with a regular partner.
Chlamydia can lie dormant for decades. There's no conclusive proof that he's caught it from his recent activities. Also, the tests throw up false positives. Get checked again. Don't try to covertly dose her, that a fucking stupid idea. Giving meds to another person without their consent is assault, and if she catches him, she'll know that he's guilty.
If he is positive for it, he needs to talk to her but gets his facts and story straight first. Whatever he does, don't ignore it because it could cause her permanent long-term harm.