How not to have SQL Server to restart on the same node but failover on 1st failure in a cluster
By default SQL Server tries to restart on the same node in case of a failure, next it does failover.


Yes correct. Just to add, in Windows Server 2003 WSFC would attempt 3 times to bring resource online(By default) but from Windows Server 2008 it would just attempt 1 times( By default). Of course you can change the value. What i have said above is documented in This Support Article


  is it possible to have SQL Server NOT try a restart on the same node and just failover in case of a failure? Max restarts in the specified period - Will making it zero do the needful?


Yes I believe so, but this should not be done. I must also say I have never changed this option to zero, never seen it people using zero in their cluster configuration, and dont know how cluster would behave( when multiple failover happens) in case it is set to zero. You must test this before proceeding with the change in prod. In almost all cases at least one restart is necessary to check, because a failover can cause longer downtime( depending on storage and complexity) and downtime for all resources which are part of resource group even if it is not their fault.