Probably not
As established in the comments here and more thoroughly in your other question, Croatian bura is probably an inherited word, at least as far back as Proto-Balto-Slavic, so any normal rules that apply to borrowings from Latin into Croatian (or even South Slavic) are not relevant here. Latin boreas is borrowed from Greek.
Going through the origins of both reveals that they’re not likely to be related, but it can’t be ruled out entirely.
Origin of bura
The entry from Derksen’s etymological dictionary linked to in a comment takes the Slavic word back to Proto-Balto-Slavic *bouʔro-, with no further reconstruction. The Wiktionary article writes this *baurjā, but they mean the same thing.
Wiktionary suggests cognacy with Latin furō ‘to rave’ and Proto-Germanic *buriz ‘favourable wind’ (< ‘carrying force’?), implicitly saying it comes from the Proto-Indo-European root *bʰer- ‘bear, carry’. As it happens, Latin furō is likely not cognate with *buriz, so as a whole, this is unlikely; perhaps it was meant to say fūr ‘thief’, which is.
Even so, this is less than convincing. It’s not usual for a PIE root like *bʰer- with *e as the nucleus to yield forms with *u as the nucleus. Indeed, fūr comes from earlier *fōr (with *ō, which is an expected ablaut alternation), and *buriz was almost certainly formed within Germanic based on a generalised *bur-, from the PIE zero-grade *bʰr̥- (also expected ablaut alternation), so neither of these two forms actually go back to anything containing an original *u.
Plus, if the reconstruction as *baurjā is correct, we want something that can yield PBS *au, which PIE *u does not. We’d need PIE *ou̯ for that, and if *bʰer- ~ *bʰur- is an irregular alternation in PIE, then *bʰer- ~ *bʰou̯r- would be downright bizarre. Indeed, if we take Derksen at his word precisely and reconstruct it *bouˀrjā, with the acute accent, we’d be looking for something like *bʰou̯Hr-, at which point any connection with *bʰer- can be safely thrown out the window.
So the default position at this point would be that Germanic *buriz is from *bʰer- ‘carry’, while Latin furō and Slavic *bùŗa are both of uncertain origin, but probably not related to *buriz.
A possible connection after all?
If we did want to connect all three words to a single PIE root, though, this would be doable. If we go ahead and blindly reconstruct a root that could potentially yield the forms we have, we would expect it to look like *bʰeu̯r- ~ *bʰur- (full grade ~ zero-grade) or *bʰeHu̯r- ~ *bʰHur-. (It could theoretically also be *bʰeu̯Hr-, but that would be a most unusual root structure, so let’s ignore that.)
The o-grade of *bʰeu̯r- would yield PBS *baur- with no acute, so that would be less than ideal; but *bʰeHu̯r- could work reasonably well:
| PIE |
Result |
| *bʰóHu̯r-i̯eh₂ |
PBS *baˀurjā |
| *bʰHur-is |
PG *buriz |
| *bʰHur-e- |
Lat. furō |
We would actually expect metathesis in PIE in the zero-grade here (*bʰHur- > *bʰuHr-), which would yield long *būr- and fūr- in PG and Latin, so it wouldn’t be quite as neat as that, but the derivation itself is not insurmountably difficult, all things considered.
More problematic is the fact that – as far as I know – the root is otherwise not attested at all. Not surprising, since I just made it up to make it fit the attested forms. To establish an Indo-European pedigree, we would prefer to see the root pop up in more than just a single derivation in Balto-Slavic, Germanic and Italic, all seemingly unrelated to each other.
All in all, the proximity of meaning between at least the Slavic and the Germanic words is noteworthy, but actually connecting them to a single, established predecessor is difficult, and our default position from above is probably still the best bet: *bùŗa is of uncertain origin.
Origin of boreas
Latin boreas is borrowed from Greek βορέας, which annoyingly is of uncertain origin. In his usual, wavering style, Beekes writes:
< IE? *gʷ(o)rH- ‘mountain’.
## Uncertain. Taken as ‘wind from the mountains’, related to a word for ‘mountain’ seen in Skt. girí-, Av. gairi-, Lith. gìrė ‘wood’, and OCS gora; cf. Illyr. bora ‘mountain’ in names (Krahe IF 57 (1940): 125ff.), as well as δειράς. So the Ὑπερ-βόρεοι are properly ‘those living beyond the mountains’ (Pedersen KZ 36 (1900): 319). The formation, however, is unclear; see Pedersen 1926: 66, Schwyzer: 461. On wind names in -ίας, see Chantraine 1933: 95. Is the word an IE formation at all?
If this origin has any truth to it, βορέας is related to Croatian gora ‘hill’, not bura ‘northern wind’. But it’s quite tentative, and there may not actually be any truth to it.
A possible connection after all?
If we assume that βορέας is not from the *gʷerH- root referring to mountains, but we still want to connect it to Slavic *bùŗa, we’re left with that eternal last-ditch option: substrate.
The two words are very similar in form and especially in the very specific meaning ‘northern wind’, that much is true, and substrate words do often display irregular patterns of sound correspondences, so it would not be wholly outlandish to suggest that (Pre-)Proto-Greek and Proto-Balto-Slavic both came into contact with some substrate language whose word for the northern wind was something like **bo(u)rea- or **bo(u)rja-, and both borrowed it.
Such substrate accounts should be taken with a stroke-inducing amount of salt, however. As this example shows, the very nature of having an unknown donor language means you can make anything fit anything, so it’s essentially a ‘free’ hypothesis, and unless you can find more data to support it, it’s worth very little.
So once again, I’d say the null hypothesis is the safest bet: Croatian bura and Latin boreas are in all likelihood not related. On the other hand, Latin was undoubtedly an influential status language in the area, and it is notable that the specific meaning ‘north(east) wind’ appears to be limited to the southern parts of the Slavosphere which had a larger presence of Romance-speakers. I would say it is fairly likely that the Latin word influenced the Slavic word in the south, nudging its meaning from the more general ‘storm’ to ‘north(east)ern [storm] wind’.
