Is it possible that the immigration officer would ask for IELTS test report even if I produce the certificate from my college/university confirming that my education was taught entirely in English?
INZ reserve the right to require IELTS if there is any doubt in a particular case. If your letter is from a reputable university, though, and you manage well at your interview, there should not be any doubt, so you should not need IELTS. And yes, this would be the same situation for your partner.
I remember hearing of someone with a university letter saying his course had been run in English who had been asked for IELTS after he did his interview with the CO. He did admit that he had had some difficulty understand the questions, and in making himself understood. His posts on the forum had always been difficult to understand, also. So you can understand that would make the CO suspicious. But they do not ask for IELTS without some good and obvious reason like that.