You need two rules for what you're trying to achieve.
The first one, the one that will receive requests and "translate" them to your under-the-hood script, should look like this:
rewrite ^/api\?(.+)$ /index.php?$1 last;
As for the second one, the one that should redirect all your users to the "beautiful" URL:
rewrite ^/index.php\?(.*)$ /api?$1 permanent;
Note that this second rule should be outside any location block and before any of those, as you're willing to redirect the user before anything else.
Cheers