Can I carry out arbitrage trades between two exchanges?

Yes, but only with shares you already hold. An Intraday buy on NSE can't be squared off on BSE, or the other way round. So cross-exchange arbitrage works only by selling shares you already hold on one exchange while buying on the other. Be careful: brokerage, exchange charges and taxes on both legs usually swallow the tiny price gap. See what is arbitrage? for the mechanics and how exchange interoperability smooths cross-exchange settlement.