Here the 1841 British edition replaced the preposition unto with into. The subsequent LDS edition (1849) restored the earlier preposition. There is a possibility that into is actually correct. Elsewhere, when the text refers to fleeing to a place, the preposition is either into (19 times) or to (13 times). There are no other instances of “fleeing unto a place”, although the use of unto in this phraseology should be possible since unto and to are basically synonymous.
There is evidence that “fleeing into a place” was once changed to “fleeing unto a place”, but only momentarily:
𝓞 is extant in 1 Nephi 5:8 and reads into. Moreover, there are 14 additional occurrences of “fleeing into the wilderness”. (There is also one occurrence of “fleeing to the wilderness”, in Ether 14:14: “they fled again to the wilderness of Akish”.) It is therefore possible that here in Alma 21:13, an original into could have been mistakenly changed to unto during the early transmission of the text. And there are quite a few cases in the history of the text where the prepositions unto and into have been mixed up; for some statistics, see under 1 Nephi 7:2.
Ultimately, here in Alma 21:13, the safest solution is to accept unto, the earliest reading, simply because there are examples of “fleeing to a place” and unto is synonymous with to. In addition, as David Calabro points out (personal communication), there is an excellent example in the King James Bible of the phraseology “to flee unto a region ... round about”:
Such biblical usage argues that unto is perfectly acceptable in Alma 21:13.
Summary: Maintain in Alma 21:13 the preposition unto in the phrase “unto the regions round about”, the reading of the earliest textual sources; nonetheless, there is a chance that this phrase originally read “into the regions round about”.