Escape ;(semicolon) in aspnet core connection string in appsettings.json file

That's not appsettings.json issue, it's connection string issue.

The basic format of a connection string includes a series of keyword/value pairs separated by semicolons. The equal sign (=) connects each keyword and its value. To include values that contain a semicolon, single-quote character, or double-quote character, the value must be enclosed in double quotation marks
from here

So your connection string shoud look like:

Server=server name; Database=my db; User Id=myuser; Password="HjkT;tk-k@=?55\"; MultipleActiveResultSets=true;

And as soon as quote and backslash are special symbols for json - you should escape them with backslash. Here is your file:

{
  "DefaultConnection": "Server=server name; Database=my db; User Id=myuser; Password=\"HjkT;tk-k@=?55\\\"; MultipleActiveResultSets=true;"
}