How can I list all flight numbers that connect two countries (non-stop)?
The Flightradar app allows you to search by route.
Searching for the route VVI (Bolivia Viru Viru International Airport) to ASU (Paraguay Silvio Pettirossi International Airport) brings up three flight numbers.
Some of these flights go to MVD airport but there is a direct flight every day.
There are also direct flights every day.
Alternative routes can be found on Skyscanner. You may have to play around with the dates to get two direct flights. If you search the flight numbers and you know what dates the flights are you can get a price of around £355 return depending on the dates:
In Skyscanner you can search for specific countries, for example Bolivia to Paraugay:
All of these methods are only practical if there are really very few flights between the two countries. So if you want a list of all flight numbers between the USA and Canada, you'll have to come up with something yourself :)
On the flight search engine kiwi.com, you can enter entire countries as both origin and destination, and you can specifically look for non-stop flights. If you then restrict your date range to maybe one week at a time (in order not to be shown the same flight dozens of times, just on different days), you should get a pretty accurate list of flights between the two countries.
If you have an idea what the origin airport might be (e. g. the largest airport in the origin country), flightconnections.com can display all routes from that airport on a map, allowing you to see the routes to the country you want. After clicking on a destination airport, hovering over a date in the calendar on the left will show you all flights on that date.
Finally, adding on to Daniil's answer, if you already know that there is only one route between the two countries, you don't have to go to Flightradar24 to see the corresponding flight number. Just enter "flight time XXX YYY" on Google, where XXX and YYY are the codes of the origin and destination airport. A list of the flights will be at the top of your search results. In this particular case however, the flight numbers and timings given by Google do not seem to agree entirely with the airline's website.
In this case, it's easy since Paraguay is small and apparently has only one airport that serves international commercial flights. https://en.wikipedia.org/wiki/List_of_airports_in_Paraguay
The easiest way would be to look at the map in flightconnections.com and click on the airports that show up. In Paraguay that's only for and only Asuncion (ASU) is the only one that connects outside the country. Click on that and you get a convenient map of all non-stop routes from ASU inclucing one to Santa Cruz, Bolovia (VVI) which is currently served by Amaszonas and Fly Africa.
For two larger countries, that search would be rather difficult. You probably would have to work your way through the larger hub airports one at a time.