You can not track external website from within qualtrics. You'll have to make changes to the destination website according.
You'll have to do some custom programming but only a subset of what you want can be tracked, monitor which website they clicked on and see how much time they spent there (this may not always be accurate).
You maybe able to incorporate the website in an iframe and then use some tools to monitor their activity within it, however, this will not be close to the real world experience you are looking for.

Thanks for your quick reply and the information.
What about tracking the time spent on one specific question (and its content) within the survey? Would there be a possibility to set up different question including some text and looking at what choices people clicked at first and how much time they spent on a specific question (text)?

Take a look at timing questions, they'll track how much time one spends on a question, when did they first click etc. There are also other questions such Heat Map that may allow you to see where people clicked.