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You certainly can!
Simply contact Boost using their live chat function available at http://livechat.boost.com.au/BOOSTorting. They are available 24/7. They will be able to assist you with transferring your number.
If chatting online isn't your thing, you can also contact Boost by calling 125 8881 directly from your new Boost SIM. This number will work even though your SIM hasn't been activated.