Yes- if you distribute through Qualtrics you can download the distribution history to see details such as if the e-mail was opened or if the survey was started (which is a link click and survey load - technically the link click itself is not measurable, this is basically the same thing).
If you opt to distribute personal links yourself, you can also tell if a survey was started upon closing the project - it will say “survey started” on that row along with any partial answers recieved.
Responses in Progress (qualtrics.com)
Responses in Progress in which the respondent’s progress is 0% are not saved by Qualtrics when they are closed.
However, you could make the first question of your survey a hidden JS click next page question, or a timing auto-next page question. That way the respondent will be automatically moved onto the next page of the survey. So, if they do then close their browser, there will still be progress made and they will be saved as a response-in-progress.