No: the apostrophe signifies the missing letter(s), as in "does not" becomes "doesn't", or "cannot" becomes "can't".
If you always used an apostrophe to signify missing letters, then typo would be typo', except that it isn't as that would be silly.
Since typo is the abbreviation for
typographical errors, the plural is clearly typos. (Of course, if typo was a word in its own right the plural would be
typoes; fortunately, it isn't

).
Had your typographical error possessed something, that would be a different matter...
(ps. You could have saved yourself some bother by using 'mistake'

)