Unable to insert nonenglish letter in sql

@vc_firstname_on_credit_card is already declared as nvarchar(50)
so you don't need to write 'N'+@vc_firstname_on_credit_card. Just use :

INSERT INTO dbo.client_billing_address (vc_firstname_on_credit_card )
                VALUES(@vc_firstname_on_credit_card)

When dealing with Unicode string constants in SQL Server you must precede all Unicode strings with a capital letter N, as documented in the SQL Server Books Online topic "Using Unicode Data". The "N" prefix stands for National Language in the SQL-92 standard, and must be uppercase. If you do not prefix a Unicode string constant with N, SQL Server will convert it to the non-Unicode code page of the current database before it uses the string.
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NVarchar is used for Unicode. If your database is not storing multilingual Data you can keep using Varchar. As an example: N'abc' simply converts your string to unicode.

change your sp as:

CREATE PROCEDURE [dbo].[client_billing_address_sp_update] 
(
@in_type tinyint,
@vc_firstname_on_credit_card nvarchar(50) = ''
) 
AS
declare @sql1 nvarchar(max)  
IF(@in_type = 1) 
BEGIN
set @sql1='insert into FCM_Claims(vc_firstname_on_credit_card) values(N'''+ @vc_firstname_on_credit_card  + ''')';
END 
exec(@sql1);

Try this function from your cs file

public Void InsertNonEnglish(string in_type,string vc_firstname_on_credit_card){
string  ConnectionString = System.Configuration.ConfigurationManager.ConnectionStrings["ConnectionString"].ToString();
      
        SqlConnection con = null;
        try
        {
            con = new SqlConnection(ConnectionString);
            SqlCommand cmd = new SqlCommand("client_billing_address_sp_update", con);
            cmd.CommandType = CommandType.StoredProcedure;
            SqlParameter[] prm=new SqlParameter[2];

            cmd.Parameters.Add("@in_type ", SqlDbType.TinyInt).Value = in_type;
            cmd.Parameters.Add("@vc_firstname_on_credit_card ", SqlDbType.NVarChar, 1024).Value = vc_firstname_on_credit_card;
        
            con.Open();
            cmd.ExecuteNonQuery();
        }
        catch (Exception ex)
        {
            throw ex;
        }
        finally
        {
            if (con != null)
            {
                con.Close();
                con.Dispose();
            }
        }
     
}