Are you seeing any "problems"?  The encoding of a column and the encoding in the client do not have to be the same; MySQL will convert on the fly.
If you want to add Asian characters or Emoji, then you do need utf8mb4 (or at least utf8).  Before embarking on a conversion, please discuss the scope of the problem.  In particular, if you are seeing 'garbage', look at Trouble with UTF-8 characters; what I see is not what I stored for diagnosing the problem.
Assuming you have correctly encoded data in your table,
ALTER TABLE t CONVERT TO CHARACTER SET utf8mb4 COLLATE utf8mb4_unicode_520_ci;
Here are 7 possible cases; you seem to have the first case:  http://mysql.rjweb.org/doc.php/charcoll#fixes_for_various_cases
If you have already stored, say, Emoji in the latin1 column, you have a worse mess; we need to discuss further.
Terminology:
- The CHARACTER SETis what bits are used to encode characters.  Eg: latin1, utf8, utf8mb4.
- The COLLATIONis how characters are compared, as forWHEREandORDER BY.  Eg: latin1_general_ci, utf8_general_ci, utf8mb4_unicode_520_ci`.
latin1 can handle all Western European characters.
utf8 can handle most characters worldwide.
utf8mb4 is a superset of utf8, and is needed for Emoji and some Chinese characters.