Can I send a British bank sterling cheque to a relative in auatralia?

You can - though there will be a fairly large charge for converting it to dollars, so unless it is a fairly large amount she may not end up with much.

It might be easier to get her account details and simply make a bank transfer - then you pay the charges and she gets the amount you want her to receive. Alternatively, order a gift card online in Australia, and pay for it by credit card

You would do better to send her some dollars with a card, and they are easy enough to get from Tesco or Western Union. If you send a cheque she will lose money when it it exchanged and run through her bank.

You can transfer money to her via Western Union as well