Yes, Google provided an API for draw routes on the map. Find below link
https://developers.google.com/maps/documentation/ios/shapes?utm_source=welovemapsdevelopers&utm_campaign=GM-for-iOS-sdk-1-7.
Here you can use Polyline to draw a route on the google map.
GMSMutablePath *path = [GMSMutablePath path];
[path addCoordinate:CLLocationCoordinate2DMake(-33.85, 151.20)];
[path addCoordinate:CLLocationCoordinate2DMake(-33.70, 151.40)];
[path addCoordinate:CLLocationCoordinate2DMake(-33.73, 151.41)];
GMSPolyline *polyline = [GMSPolyline polylineWithPath:path];
polyline.map = mapView_;
To enable the poly line you have create a Direction Api Key for your project. Using that api key you can draw the poly line.
I am sharing an answer link below.
Getting status "REQUEST_DENIED" after fetching data from google places API.
In the above answer they mentioned some other api. In your case it is Direction api.