I am a Ph.D. student in my third year. I was hoping to find an opportunity to do visit research in some other universities. Are there any suggestions on the way to find an opportunity?
As a student, I'd think there are few opportunities. Personal contact with someone you want to work with is probably the best way to go about it, but blind emails to professors are often ignored. As a student you are asking for something, but may not have a lot to offer in return.
But, if you can get your supervisor/advisor to make enquiries on your behalf then can be possible. It can be a way to initiate collaborations that are mutually advantageous. And, make sure that you know something about anyone you approach so that you are sure your research interests align with theirs.
Funding will also be an issue and you need to resolve that separately. That might be the biggest obstacle unless you have something very important to contribute to someone else's research. That is more likely for professors than students.