You can build a custom Apex REST web service, and call that to convert your Lead:
https://success.salesforce.com/answers?id=90630000000hLeeAAE
Then build an SFRestRequest and set it's endpoint to your service
SFRestRequest *request = [SFRestRequest requestWithMethod:SFRestMethodPOST path:@"" queryParams:nil];
request.endpoint = @"/services/apexrest/{your endpoint}/{a lead Id}";
[[SFRestAPI sharedInstance] sendRESTRequest:request failBlock:^(NSError *e){
NSLog(@"error: %@", e);
} completeBlock:^(id success){
NSLog(@"success: %@", success);
}];
convertLead() is available in the SOAP API. I don't think it is possible to call convertLead directly from the REST API. If you'd like to use the SOAP API in your iOS application, use this library: