This is as much a question about regular expressions (regex) than .htaccess. The second argument (CondPattern) to the RewriteCond directive takes a regex, not a wildcard expression. (Specifically, Apache uses the PCRE flavour of regex.)
Would that just be RewriteCond %{QUERY_STRING} ^m=*$
No. This assumes the argument is a simpler wildcard type expression (where * simply matches 0 or more characters) - it is not. In regex-speak, the * matches 0 or more occurrences of the preceding character/group. So, this would match exactly ?m (0 occurrences), ?m= (1 occurance), ?m== or ?m===, etc. Not your intention at all. It wouldn't even match your original URL.
Assuming the m URL parameter always occurs at the start of the query string, then to match ?m=<anything>, you could simply use the regex ^m=. This matches any query string that starts m= followed by anything (including nothing). For example:
RewriteCond %{QUERY_STRING} ^m=
However, you generally want regex to be as restrictive as possible (although that may not apply in this case). For example, to match any URL that contains the query string ?m=N only, where N is a digit 0-9 and there are no other URL parameters, then you could do something like:
RewriteCond %{QUERY_STRING} ^m=\d$
Where \d is a shorthand character class for any digit (the same as [0-9]).
Aside: Test with 302 (temporary) redirects to avoid caching problems and clear your browser cache before testing.