Google is not the all seeing omnipresent Goomba of the Internet. How do you suppose they will know what happens on your site once a visitor gets there? Are you making the assumption that Google Analytics (GA) is a Google Search spy?
Seriously, people give Google far too much power even if it is sometimes only in their mind. Google will not know that someone has left your site unless they back-arrow to the SERPs.
Now onto something far more serious. If you want people to come back to your site, do not make them feel burned. Pop-ups for conversion when a user lands on a page is a sure fire way of getting that famed back-arrow. It is kinda like a particular finger in the air.
If you want to effect bounce rate, do not look to tricks, but do honest hard work and attract the exact right searcher for your page. Keep it simple. Conversion only happens when the user is happy and not deceived.
To answer your question, Google will only count a bounce if it is a bounce. What you described is not a bounce. It is an exit.
Okay. For Google Search and Google Analytics, you have to think differently. For GA, here is how Google defines bounce and exit according to this link.
1] For all pageviews to the page, the Exit rate is the percentage that were the last in the session.
2] For all sessions that start with the page, Bounce rate is the percentage that were the only one of the session.
3] The Bounce rate calculation for a page is based only on sessions that start with that page.
If that is not confusing, I am not sure what is. Read the linked page and you will see that Google's definition of a bounce does not fit the rest of the world.
Where with Google Search, bounce as defined above cannot be determined and often just means a return to the SERPs within a short period of time sometimes defined as less than 10 seconds.
Using your example, in GA a click on your pop-up would be a bounce though I am not sure that would be fair. It would result in a high bounce rate (assuming I read their example correctly). There is a lot of discussion as to how Google defines a bounce in search and GA where it is clear that the GA bounce rate is a poor indicator of performance. However, the same discussion regarding bounce rate in search seems to be more nuanced allowing it to be a factor in a set of metrics where an extremely high bounce rate can indicate high quality over a very low bounce rate where it is assumed that the information sought was not found on the landing page.