Sure having K2 talk to both SharePoint 2010 and 2013 is not a problem.  K2 has some documentation for configuring K2 with SharePoint 2013 that you'll want to review and follow.

K2 for SharePoint Product Documentation: Installation and Configuration Guide

http://help.k2.com/onlinehelp/K2ForSharePoint/ICG/4.6.11/default.htm

I'd recommend first off making sure your SharePoint 2013 environment is setup correctly for Apps including the App Catalog and the User Profile Service.  A good test before trying to configure K2 is try and deploy one of the free SharePoint Store apps, like Sticky Note, and make sure that works.  If that doesn't work, then the K2 App will not work either.  This will help focus your troubleshooting on either SharePoint and/or K2. The below link discusses some of those prerequisites for SharePoint on premise. 

Preparing SharePoint On-Premises

http://help.k2.com/onlinehelp/K2ForSharePoint/ICG/4.6.11/default.htm#Install_Topics/Before/SP_Requirements.htm#SharePoint

Deploy the K2 for SharePoint App to the App Catalog

http://help.k2.com/onlinehelp/K2ForSharePoint/ICG/4.6.11/default.htm#Install_Topics/OnPrem/Deploy_App.htm%3FTocPath%3DInstall%7CSharePoint%25202013%2520(On-Prem)%2520Installation%7C_____2

Also note that the K2 App Deployment tool will also do some checks before trying to deploy the app and will let you know if there are problems with some of the prerequisites. 

Hi Tim,

Thanks for the response.

Will this one be applied to have two connections i.e. SharePoint 2010 and SharePoint 2013 to connect to same K2 DB and K2 server at a same time.

Actually we are process to migrate the SharePoint 2010 K2 to SharePoint 2013 K2, hence we decided for any kind of option to make a connection of SharePoint 2013 to K2 DB and server without disconnecting SharePoint 2010 to the same K2 DB and server.