I am using the following SQL command in my linked MS SQL Server :

SELECT * Into #LastMonth FROM OPENQUERY(QODBC, 'sp_report CheckDetail show TxnID,TxnNumber,SplitAccount,ClearedStatus, TxnType, RefNumber as CheckNumber, Date as CheckDate, Name, AccountNumber,Account, PaidAmount, OriginalAmount parameters DateMacro = ''LastMonth''')

This takes 2-4 minutes to run and returns 1500-3000 records. How can I make it faster?

Unfortunately SP_REPORTS are not items that can be optimized with the QODBC v6 Optimizer. As long as you have the ODBC Provider Options > Allow InProcess proptery set:-

(setup during your setup of the link server) I think that is the best you can do. QuickBooks reporting does not perform well when threaded which is how SQL calls QODBC.