Like you say, you'll want to escape quote characters.
See this documentation on string literals:
There are several ways to include quote characters within a string:
A “'” inside a string quoted with “'” may be written as “''”.
A “"” inside a string quoted with “"” may be written as “""”.
Precede the quote character by an escape character (“\”).
A “'” inside a string quoted with “"” needs no special treatment and need not be doubled or escaped. In the same way, “"” inside a string quoted with “'” needs no special treatment.
Depending on how you're dealing with the SQL, though, you may need to do more than that. If the application is escaping the quote character, and passing that to a stored procedure call, you may run into the same issue if you are not using parameter binding with prepared statements. This is due to MySQL removing the escape character upon processing the inputs of the SP. Then the unsantized character makes its way to the query construction and the problem repeats itself if it should be escaped there. In this case, you'll want to switch to parameter binding, so that the escaping and query construction is out of your hands.