Short answer
- To escape single quote / apostrophe, embrace the string containing an apostrophe between double quotes (").
- To escape double quotes, apply a double substitution, first to remove the double-quotes, then to add them again.
Explanation
Google Sheets QUERY built-in function automatically escape some characters by internally adding \ before single quotes but it's doesn't work when the cell value to be used as the source for the criteria includes double quotes. As a workaround, the the use of double substitution is proposed.
Example for single quote / apostrophe
Below table represents and spreadsheet range that contains
- Column A: The data source
- Cell B1:  The data value to be used in the criteria expression
- Cell C1: The following formula =QUERY(A:A,"SELECT * WHERE A = """&B1&""" ")
+---+---------+-----+-----+
|   |    A    |  B  |  C  |
+---+---------+-----+-----+
| 1 | I'm     | I'm | I'm |
| 2 | You're  |     |     |
| 3 | It's    |     |     |
| 4 | I am    |     |     |
| 5 | You are |     |     |
| 6 | It is   |     |     |
+---+---------+-----+-----+
Example for single quote / apostrophe and double quotes
=SUBSTITUTE(
   QUERY(
     SUBSTITUTE(A:A,"""","''"),
     "SELECT * WHERE Col1 = """&SUBSTITUTE(B1,"""","''")&""""
    ),
    "''",""""
 )
Note that instead of using a the letter A as identifier of the data source column it's used Col1.
Reference
https://developers.google.com/chart/interactive/docs/querylanguage