Sarah – whilst your Landlord is required to ensure that the premises is maintained in a state of “reasonable repair”, in practical terms that really applies to habitable areas, not the garage.
Nonetheless the possible presence of asbestos containing residues in the garage renders that area unusable, and the environmental testing and/or remediation of that area is, once you formally advise them, a matter for your Landlord to arrange.
If by some chance the “garage” that you’re referring to is actually carspace, then then parts of it may be the Common Property of the Owners Corporation (O/C), but even then the environmental testing and/or remediation works that you’re requesting would be a matter between your Landlord and their O/C – nothing to do with you and nothing for you to pay for.
So if you haven’t done so already, you need to write to your Property Manager (Rental Agent) advising them of your concerns, of your consequent unwillingness to use the garage (that you’re paying for), and giving them say 7 days to commission appropriate environmental testing as a matter of urgency, say a further 14 days to attend to any remedial works and/or to clean-up works (depending upon the test results), and advising that should they not meet your timeframe for testing then you will arrange for that and, irrespective of the result/s, seek a full reimbursement of your costs up to the O/C’s quoted $500 per test from the Landlord and in default by seeking Orders in that regard, including for economic loss (e.g. paying or a garage that you’re unable to use) from the NSW Civil and Administrative Tribunal.
Finally, the amount quoted by your Landlord’s O/C is excessive, so as a contingency, contact Safe Environments (https://www.safeenvironments.com.au/), they’re very helpful and on the last occasion that I used them (2013) charged $90 for the asbestos testing of each sample delivered to their Blacktown NSW premises.