Can I disclose a conflict of interest with some professors from a particular university in responding to comments?

I'd advise against such a move, which can be seen as a way to avoid a complex revision.

Usually, at the time of submission, journals allow the authors to specify non-preferred potential reviewers, either directly through the submission management system or through the cover letter. If you think that someone can be strongly biased against your work due to a conflict of interest, that's the time to voice your concern, not after the review.

I would not do something like that and would similarly discourage you from doing that. There are several reasons why you should not do so:

1) You are assuming that you know who the person is. While you may have some confidence - there remains a probability that the reviewer is not who you have in mind.

2) You are also assuming that the reviewer knows your identity even though I am assuming it is a double blind review.

3) There is a further assumption that he/she will act against you deliberately because he has a bone of contention with your supervisor.

I know academics can be catty but I think you can trust the journal process and academics generally, to be impartial (generally speaking).

But suddenly, you somehow guess with a high probability that a person who is a long time adversary of your supervisor may be behind this comments and he won't be satisfied no matter what you do.

You don't need to satisfy the reviewer. Obviously it makes life much easier if you can get the reviewer's endorsement, but ultimately the person you need to satisfy is the editor.

If the reviewer has requested revisions, and you can satisfy the editor that you've done everything they asked for, then it's going to be hard for the reviewer to argue against acceptance. My recommendations would be:

1. Seek clarification on anything ambiguous in the reviewer's comments. (I recently got one that simply said "the graphics are very poor and need to be improved" - rather than spend a long time trying to guess what the reviewer wanted, I asked the editors to clarify what was needed.)
2. Where you can reasonably do so, make the edits that the reviewer has requested.
3. Produce an itemised response that lists what you've done in response to each of the points they raised. If you can show that you've addressed all the points they made, it's then very hard for them to argue against acceptance. Even in a non-hostile situation this is a courtesy to reviewers and editors since it helps them understand how you've changed the document.
4. If you disagree with some of their recommendations, look for non-adversarial ways to frame that.

Number 4 there is complex, so let me unpack that a bit. The idea is not to say "you're wrong so I'm going to ignore this", but rather "I can't respond to this issue in the way you recommended, but here's how I can respond to it".

For instance, in my recent paper I proposed a model that involved a set of random variables with distribution N(0,theta^2). One of the reviewers said something along the lines of "presumably this should be N(1,theta^2)?"

My initial reaction was "no, that's wrong, you've misread it" - and they had misread it, 0 was definitely the correct value. But what I ended up going with was "I've modified the form of the model to clarify why N(0,theta^2) is used instead of N(1, theta^2)".

Part of my reason for making this change was that the reviewer comment showed me that my working wasn't as clear as it could be. But it was also a diplomatic decision - I wanted to show that I was willing to take their feedback into account, even when I couldn't implement it as suggested.

If you're lucky, this sort of approach may help disarm possible reviewer hostility. But even if it doesn't, it shows the editor that you're receptive to comments, and minimises the grounds on which the reviewer can object to your paper.