I need to create a query as following.

$user = User::select("users.*",DB::raw("CONCAT(users.first_name,' ',users.last_name) as full_name" ->get();

and then I need to use the full_name on an advanced where clause.

I tried above laraval code and doesn't worked at all. Is there are any method to do this without using raw queries ?

Try this

User::select('*')->addSelect(Db::raw("CONCAT(first_name,' ',last_name) as full_name"))->whereRaw("CONCAT(first_name,' ',last_name) = 'Full Name'")->get();

That's right, I'm not sure you can query these computed columns in your where SQL statement, so you have to compute them again in the query, as flynsarmy pointed out