we need to do upgrade SharePoint 2010 farm to SharePoint 2013,
Does SharePoint 2013 supports SQL Server 2016 or not? or we can go for Server 2014?
SharePoint 2013 will never be supported on SQL 2016. (with the exception of SSRS integration, keep reading for that).
SharePoint support generally works in the following way. A SharePoint release will support the latest SQL Server version(n) at the time of release as well as the one previous released version (n-1) and one addition/new release of SQL (n+1). For the case of SharePoint 2013, the follow SQL versions are supported:
SQL 2014 - n+1 SQL 2012 - n SQL 2008 R2 - n-1
According to the following link, the answer is no, SQL Server 2016 is not supported with SharePoint 2013.
SQL Server 2014 is supported, but should have the May 2014 Cumulative Update installed