Till now I have been into developing SharePoint Add-ins, custom web pages, etc. My only query is with SharePoint more and more towards client side, Can I use SPFx in SharePoint 2013 online Office 365 version? An answer would be highly appreciated.
i am confused to what your referring to "SharePoint 2013 online Office 365 version"
SharePoint 2013 is a separate version to SharePoint Online and Office 365.
within SharePoint Online SPFx was created mainly due to the fact that it was difficult to create webparts and features that ran like the older versions (fully trusted deployed to server version for on-premises releases) like we had in 2013 and less.
the other issue was getting sharepoint inline with current web standards and having the possibility to make it more client side rather than server side code which was sluggish.
so, short answer is yes its 100% possible in SharePoint online and within office 356 within the new look and feel (modern experiences) to create SPFx applications. Its possible to do it in SharePoint 2016. not possible in 2013 and dont think that they would be updating it so that it is.