Quickly is used when something happens in a short time interval. The car went from place A to place B very quickly. I don't see any example of a car going quickly in ngrams https://books.google.com/ngrams/graph?content=car+is+going+quickly%2C+car+is+going+fast
I (native AmE) recommend the addition of "quickly" as an acceptable translation, although I would also say that "fast" would likely be used much more often, at least in the US.
In the Cambridge Dictionary (https://dictionary.cambridge.org/dictionary/english/fast) the definition provided for "fast" as an adverb is... "quickly." The first example sentence given under that definition is "The accident was caused by people driving too fast in bad conditions." Additionally, the first definition in the same source (https://dictionary.cambridge.org/dictionary/english/quickly) for “quickly” is...”at a fast speed,” giving as an example sentence, “The plane climbed quickly to a height of 30 000 feet.”
So, since the adverb “fast” is defined as "quickly," and the adverb “quickly” is defined as “at a fast speed,” and since usage examples for both definitions include sentences that reference movement of a vehicle, I will be happy to add translations that use "quickly" here, if the CZ natives on the team agree.
Not just in the US it seems. Dictionary examples, even when quoted fairly, can mislead the language student. And IMO here it is extra subtle.
The plane's climb is very likely viewed as a completed event when you say "climbed quickly". With your knowledge of Czech, you might be feeling the aspect difference even in English, even though native English speakers usually do not stop to think about aspect hiding in their simple past usage.
Why is aspect, a category mostly hidden in English, relevant? Somehow the imperfective makes "quickly" sound weird. It is all fine when native speakers exercise their God-given right to sound weird, but non-native speakers are often held to a higher standard and might want to choose their adverbs wisely.
There is a way of getting insights into this from ngrams based on the pre-emption of perfectiveness by using the past progressive. Notice how marginal "quickly" gets when combined with the progressive here. While the ratio of "fast" to "quickly" is only about 2.0 with the simple past, it shoots up "quickly" once we force the imperfective context. Presumably this frequency shift is an indication that plenty of native English speakers feel the imperfective awkwardness of "quickly" just like this ESL speaker.
Combining the weird "quickly" with the equally weird simple present "goes" is just a bridge extra far. If a car goes quickly, I mostly see it relieving itself promptly. We should consider the impact of accepted translations on the learner of our native languages. If we had the MARGINAL flag to alert the learner, many of these endless arguments would go away. We probably never will.
Although my automatic choice for this sentence would be “fast,” I do feel that “fast” and “quickly” are often interchangeable. But...
Overnight, I began to doubt my “logic” of yesterday. As I toyed with sample sentences of my own, usually both “fast” and "quickly" sounded fine. But in some cases, “quickly” somehow felt wrong, though I couldn’t quite say why. (There’s that “speaking X since forever” thing again...) But your bringing subtlety, aspect, and tenses into the discussion may have shed some light on this for me, and possibly for others as well.
I have no big bone to pick (haha) with a decision against accepting "quickly" here; the issue really needs to be put to rest and you’ve provided the needed inducement for its well-deserved slumber. As always, I appreciate not only your insightful response, but also the enormous amount of time and energy that you and the other “lifers” have put into creating this course and making it work.
Finally, a MARGINAL flag would be a wonderful thing! Maybe in 2024...?