Does it matter to convert integer primary key into equivalent string?
There are potential performance differences and sorting differences (collation dependent) per this DBA.StackExchange answer and it's linked answers such as this one. I highly recommend reading through all the information referenced in that answer.
These sections of the above linked articles are specifically relevant:

There is no performance difference among these three types [char,
varchar, text]...


Short answer: integer is faster than varchar or text in every aspect.

Therefore integer (and BIGINT) is more performant of a data type, even of equal data length.