First.h

@interface JAHFirstViewController : UIViewController // Note, JAH class prefix, you should prefix all of your classes

@property (nonatomic, strong) NSMutableArray *json; /* this property is named poorly – it's not json, it's an array */

@end

First.m

// @synthesize json; <- this is completely unnecessary

- (void)doSomething
{
    SecondViewController *secondViewController = [[SecondViewController alloc] init];
    [secondViewController setSecondJSON:[self json]];
    [[self navigationController] pushViewController:secondViewController animated:YES];
}

Second.h

@interface JAHSecondViewController : UIViewController // Again, prefix your classes

@property (nonatomic, strong) NSMutableArray *secondJSON; // Again, this is named poorly, fix it!

@end

Second.m

- (void)viewDidLoad
{
    [super viewDidLoad];

    NSLog(@"%@", [self secondJSON]);
}