That setting does not affect the stored proc, it affects the script you are running to CREATE the proc. There is no reason to change it.
When SET QUOTED_IDENTIFIER is ON (default), identifiers can be delimited by double quotation marks (" "), and literals must be delimited by single quotation marks (' '). All strings delimited by double quotation marks are interpreted as object identifiers.
SET QUOTED_IDENTIFIER takes effect at Transact-SQL parse-time and only affects parsing, not query optimization or query execution.