No you cannot upgrade with avios, however if there is business award availability you can call to change from economy to business.
However, if you change from economy to business, you will normally be charged the change fee of £35 in addition to the extra Avios.
IME, for many redemption tickets using Avios for QF domestic flights, the cash component (taxes, fees and charges) is much less than that - typically £10 or so. If you cancel the original economy class ticket, you are refunded all of your Avios and you only forfeit the £10 or so (provided you don't cancel too late).
Therefore it's very often cheaper to make the change from economy to business by cancelling the economy class award ticket and booking a new business class ticket. In practice, of course, you should do it the other way round: book the new ticket first, then cancel the existing one - just in case something goes wrong with the new booking.