The Danish word comes from Old Norse dœgn, ultimately a derivation from Proto-Germanic dōgaz ‘daily’. (If one is familiar with comparative Germanic linguistics, the long vowel in the Old Norse form speaks against any borrowing from Latin diurnāta). Meanwhile, Latin diurnāta is a derivation from dies ‘day’, unrelated to the Germanic word.

If you are curious about where words come from, the English-language Wiktionary often has etymologies in its entries for words, and for the Nordic languages they are usually pretty up-to-date in terms of the scholarly state of the art.