Welcome to the CoI Community, Tanzila!

Yes, the diagram for the CoI framework and the CoI survey are both openly licensed. You have permission to use them, including modifying them, if you cite their sources and the specific Creative Commons license (CC-BY-SA).

If you do end up modifying them, you will also have to note in your citation how they have been modified from the original. Note that, by the Share-Alike (SA) rules in the license, you would then have to license your modified version under the same license, allowing others to use it as well. (There is no central registry for Creative Commons licensing; just include the license - CC-BY-SA - in your citation.)

If your dissertation or other research is focused on the CoI, let us know when you're done; there is always room in the CoI Repository for dissertations and other graduate work.

I want to use the teaching presence section of the questionnaire and want to modify the questions a little bit so that it records the teachers perspectives, not the students? So after doing that how to go for the licensing and all?That part is not clear to me.

The best example for the citation format is given above the downloadable CoI Framework diagrams here:
https://www.thecommunityofinquiry.org/coi

The "SA" part of the CC-BY-SA license listed there refers to a "Share-Alike" policy: if you use the diagram and modify it, you have to let others use your modified version as well - we share our version with you, so you share your version with others. To do that, all you have to do is include the same statement in your own citation, something like this:

The Community of Inquiry framework, modified. The original framework has been modified to include... [describe how you've modified it]. The original image is used with permission from the Community of Inquiry website; the modified image is licensed under the CC-BY-SA International 4.0 license (https://creativecommons.org/licenses/by-sa/4.0/). The original image is located at https://www.thecommunityofinquiry.org/coi

That statement will let anyone use your modified image, as long as they give you credit (and as long as they share their own modifications as well). By including that statement with your image, your image will be licensed, instantly and automatically.

We can't really give direct advice on your research here - your research is up to you and your advisor(s). I will simply note two strengths of the CoI Framework. One strength is that the three presences (Teaching, Social, and Cognitive) are integrated into a single educational experience, so the presences are often treated together rather than separately. A second strength is that the wording in the CoI survey has been carefully validated (the "validation studies") to demonstrate that the survey really is measuring what it claims to measure; that validation is based on the language that it has now and wouldn't cover any modifications. You would, of course, have to weigh the importance of those two strengths for the research you want to do.

Thank you very much for your helpful response.