SQL injection in Postgres functions vs prepared queries
It depends.
SQL functions
With LANGUAGE sql, the answer is generally yes.
Passed parameters are treated as values and SQL-injection is not possible - as long as you don't call unsafe functions from the body and pass parameters.
PL/pgSQL functions
With LANGUAGE plpgsql, the answer is normally yes.
However, PL/pgSQL allows for dynamic SQL where passed parameters (or parts) are concatenated to a query string and executed with EXECUTE. This can convert user input to SQL code and makes SQL injection possible. You cannot tell from outside whether the function body handles it properly. Tools are provided.
Only use dynamic SQL where you need it. Plain SQL statements using parameters as values are safe against SQL injection like SQL functions.
For dynamic SQL, preferably pass values as values with:

the USING clause. Example.

Makes SQL injection impossible on principal.
If you concatenate values in the SQL string, use:

format() with format specifier %L. Example.
quote_literal() or quote_nullable(). Example.

Wraps strings in single-quotes safely, thereby avoiding syntax errors and SQL injection.
Process parameters that shall be treated as identifiers in the SQL string with:

format() with format specifier %I. Example.
quote_ident(). Example.
a cast to a registered type - regclass for table names:  _tbl::regclass. Example.

Encloses identifier strings in double-quotes safely where required, thereby avoiding syntax errors and SQL injection.
Related:

Refactor a PL/pgSQL function to return the output of various SELECT queries
Define table and column names as arguments in a plpgsql function?

Never just build a string from user input and execute. This includes identifiers, directly passed by a user or fetched from a system catalog. Everything has to be treated like user input and quoted appropriately when building dynamic SQL!
More about performance implications in this related answer:

Function Performance

Basics on SQL-injection:

http://bobby-tables.com/

Similar considerations apply to other server-side languages that allow dynamic SQL.