I have struggled to find a contemporary account of the meeting between Brahe and Kepler that mentions the languages used, though with further research I may be able to come up with something.
The facts we do know:
Brahe obviously spoke Danish, and he also spoke Latin very well, and wrote De Nova Stelle in Latin (quite good Latin actually) Brahe also spoke German from his 8 years of Maths and Astronomy training in Germany so he was at least proficient in German. And 8 years speaking a language tends to make on fully proficient, especially in speakers that are already polyglots.
"In 1562 he travelled to Leipzig to study law, but Tycho was more interested in astronomy. He discovered large discrepancies between his own observations of the positions of the planets and the positions noted by astronomers before him. Tycho Brahe returned home from Leipzig in 1565 and between 1566 and 1570 he studied astronomy in Rostock, Wittenberg, Basel and Augsburg." link
Kepler, a German native, spoke High German. As he was on a theological tract, he wrote and spoke Latin extremely well, in fact some have said his mastery of Latin as second to none. I havent found any proof Kepler spoke Danish.
"He was educated in Swabia; firstly, at the schools Leonberg (1576), Adelberg (1584) and Maulbronn (1586); later, thanks to support for a place in the famous Tübinger Stift, at the University of Tübingen. Here, Kepler became Magister Artium (1591) before he began his studies in the Theological Faculty. At Tübingen, where he received a solid education in languages and in science" link
I would also like to note one other possibility, as both were in the Court of Prague, there would be a slim chance they learned Bohemian, but I find this the least likely of the 2, the Court in Prague also spoke Latin and German, so it would be odd to conduct highly technical scientific business in a language one is just learning, when they are already are mutually proficient in Latin and German.
"In 1597 Tycho Brahe left Denmark, and in 1598 he accepted a proposal from Emperor Rudolf II that he come to Prague, where he was to work with Johannes Kepler. Tycho Brahe died in 1601, possibly of ureamia, possibly because of mercury poisoning." link
I know this isnt the exact answer you want, but I think it is safe to assume the conversations were in either Latin or German, but I would sort of lean towards Latin as it was the language of science at the time.
"Latin was the language of international communication, scholarship and science until well into the 18th century, when vernaculars (including the Romance languages) supplanted it." link
To expand on Ed's answer: In speaking about astronomy or mathematics the two scholars would inevitably have done so in Latin. It is not only that both wrote all their books in Latin; at the time in question neither German nor Danish had the specialised technical vocabulary needed to talk about scientific matters. Latin was definitely still the language of science in Western Europe. It is obviously possible that they might have engaged in polite small-talk in some other language, but there is no evidence that they did. In the absence of evidence there is no ground for assuming that to be the case.